Abramowitz, Milton (ed.); Stegun, Irene A. (ed.) Handbook of mathematical functions with formulas, graphs, and mathematical tables. 10th printing, with corrections. (English) Zbl 0543.33001 National Bureau of Standards. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. xiv, 1046 pp. £42.70 (1972). See the review of the first edition (1964) in Zbl 0171.38503. For errata and corrections, see Van E. Wood, Math. Comput. 23, No. 106, 467 (1969); Lester Guttman, ibid. 23, No. 107, 691 (1969); Harry Björk, ibid. 23, No. 107, 691 (1969); Josef Stein, ibid. 24, No. 110, 503 (1970); Anthony J. K. Strecok, ibid. 25, No. 114, 405 (1971); S. Kölbig and F. Schäff, ibid. 26, 1029 (1972); Henry E. Fettis and James C. Caslin, ibid. 27, 219 (1973); H. H. Denman and B. D. Emaus, ibid. 27, 219–220 (1973); Robin S. McDowell, ibid. 27, 681 (1973); Ove Skovgaard, ibid. 28, 1182 (1974). Cited in 16 ReviewsCited in 2985 Documents MSC: 00A20 Dictionaries and other general reference works 00A22 Formularies 33-00 General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to special functions 65A05 Tables in numerical analysis 44A10 Laplace transform 65Dxx Numerical approximation and computational geometry (primarily algorithms) 62Q05 Statistical tables 11B68 Bernoulli and Euler numbers and polynomials 11M06 \(\zeta (s)\) and \(L(s, \chi)\) 11Y70 Values of arithmetic functions; tables Citations:Zbl 0515.33001; Zbl 0171.38503 PDF BibTeX XML Full Text: Link OpenURL Online Encyclopedia of Integer Sequences: Euler totient function phi(n): count numbers <= n and prime to n. a(n) is the number of partitions of n (the partition numbers). Powers of 2: a(n) = 2^n. Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n. Sum of proper divisors (or aliquot parts) of n: sum of divisors of n that are less than n. Numerator of Bernoulli(2*n)/(2*n). Twin primes. Primes == +-1 (mod 8). Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1). Stirling’s formula: denominators of asymptotic series for Gamma function. Lesser of twin primes. a(n) = binomial(n, floor(n/2)). Pythagorean primes: primes of form 4*k + 1. Possible values for sum of divisors of n. Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x). Largest Stirling numbers of second kind: a(n) = max_{k=1..n} S2(n,k). Primes p == +- 3 (mod 8), or, primes p such that 2 is not a square mod p. Binomial coefficient C(3n,n-1). Numbers n such that n and n+1 have the same number of divisors. Primes p such that 2p-1 is also prime. a(n) = (n-1)*n*(n+4)/6. The odd prime numbers together with 1. Balanced primes (of order one): primes which are the average of the previous prime and the following prime. a(n) = denominator of Bernoulli(2n)/(2n). Reversion of g.f. for Fibonacci numbers 1, 1, 2, 3, 5, .... Primes of form 8n+1, that is, primes congruent to 1 mod 8. Primes of the form 8k + 5. Primes of the form 8n+7, that is, primes congruent to -1 mod 8. Final digit of prime(n). Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n. Number of ways of writing n as the sum of 2 triangular numbers. Möbius (or Moebius) function mu(n). mu(1) = 1; mu(n) = (-1)^k if n is the product of k different primes; otherwise mu(n) = 0. Triangle of central factorial numbers |t(2n,2n-2k)| read by rows. Triangle of central factorial numbers |4^k t(2n+1,2n+1-2k)| read by rows (n>=0, k=0..n). a(n) = tau(tau(n)). Decimal expansion of zeta(5). Decimal expansion of zeta(16). Decimal expansion of zeta(17). Decimal expansion of zeta(18). Decimal expansion of log(4). Decimal expansion of log(11). Decimal expansion of log(12). Decimal expansion of log(13). Decimal expansion of log(17). Decimal expansion of log(20). Decimal expansion of log(21). Decimal expansion of log(26). Least integer of each prime signature A124832; also products of primorial numbers A002110. Euler’s table: triangular array T read by rows, where T(n,k) = number of partitions in which every part is <= k for 1 <= k <= n. Also number of partitions of n into at most k parts. Differences between consecutive odd primes, divided by 2. Least integer of each prime signature, in graded (reflected or not) colexicographic order of exponents. Numerators in Taylor series for x * cosec(x). Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038. Number of preferential arrangements (onto functions) associated with each numeric partition, partitions in Abramowitz and Stegun order, irregular triangle read by rows. 1 + Sum_{n >= 1} Sum_{k = 0..n-1} (-1)^n*T(n,k)*y^(2*k)*x^(2*n)/(2*n)! = JacobiCN(x,y). Triangle T(n,k) read by rows giving coefficients of Hermite polynomial of order n (n >= 0, 0 <= k <= n). Triangle in which n-th row lists all partitions of n, in graded reverse lexicographic ordering. Triangle of coefficients of Chebyshev polynomials T_{2n+1} (x). Array of number of partitions of n into m parts which have the parts of the partitions of m as exponents. Characteristic array marking partitions of m whose parts are exponents of partitions of n into m parts. Decimal expansion of ”lemniscate case”. Numbers n such that d(n) >= n-th harmonic number H(n)=sum_{i=1..n}1/i. Numbers n such that, for some numbers (j,k), j<=k, n is the smallest positive multiple of j (or more) of the first k positive integers. Numbers k such that, for all m < k, d_i(k) <= d_i(m) for i=1 to Min(d(k),d(m)), where d_i(k) denotes the i-th smallest divisor of k. Numerator of partial sums of a certain series. First member (m = 2) of a family. Array of multinomial numbers (row reversed order of table A036039). Irregular triangle T(n,m) (n >= 0) read by rows: row n lists numbers of distinct parts of partitions of n in Abramowitz-Stegun order. Semiprimes p*q where both p and q are primes of the form 6n-1 (A007528). Waring numbers for power sums functions in terms of elementary symmetric functions; irregular triangle T(n,k), read by rows, for n >= 1 and 1 <= k <= A000041(n). Parity of partitions of n, with 1 for even, 0 for odd (!). The definition follows. Expansion of elliptic modular function lambda in powers of the nome q. Array of product of parts of the partitions of n with only prime parts. Characteristic array for partitions with only prime parts. Expansion of q^2 in powers of m/16 where q is Jacobi nome and m is the parameter. Numerators of expansion of original Debye function D(3,x). Numerators of expansion for Debye function for n=1: D(1,x). Numerators of expansion for Debye function for n=2: D(2,x). Denominators of expansion for Debye function for n=2: D(2,x). Numerators of expansion of Debye function for n=4: D(4,x). Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0 <= k <= n. Triangle of coefficients of (1 - x)^n*B_n(x/(1 - x)), where B_n(x) is the n-th Bell polynomial. Triangle of coefficients of n!*(1 - x)^n*L_n(x/(1 - x)), where L_n(x) is the Laguerre polynomial. Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1} x[k]*(t^k)/k!). Monic integer version of Chebyshev T-polynomials (increasing powers). One half of even powers of 2*x in terms of Chebyshev’s T-polynomials. Coefficient table for Chebyshev polynomials T(2*n,x) (increasing even powers x, without zeros). Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of sqrt(2)/2 and 1. Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (AGM) of sqrt(3)/2 and 1. Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 1/2 and 1. Numerators of partial sums of a series for the inverse of the arithmetic-geometric mean (agM) of 2/sqrt(5) and 1. Triangle T(n,k), 0<=k<=n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1,k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles. Triangular table of denominators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x). Triangular table of coefficients of Laguerre-Sonin polynomials n!*2^n*Lag(n,x/2,1/2) of order 1/2. Triangular table of numerators of the coefficients of Laguerre-Sonin polynomials L(1/2,n,x). Irregular triangle read by rows: coefficients C(j,k) of a partition transform for direct Lagrange inversion. Triangle read by rows: row n gives coefficients of polynomial P_n(x)= U_n(X,1) + 3 * U_{n-2}(X,1) where U is the Chebyshev polynomial of the second kind, in order of decreasing exponents. Triangle read by rows: coefficients of a Bessel polynomial recursion: P(x, n) = 2*(n-1)*P(x, n - 1)/x - n*P(x, n - 2) with substitution x -> 1/y. Triangle read by rows: expansion of (1+3*t^2)/(1-t*(2*x-t)). Triangle read by rows: numerators of coefficients of the Debye-type polynomial u_n used for asymptotic Airy-type expansions of Bessel functions of arbitrary large order. Coefficients for expansion of (g(x)d/dx)^n g(x); refined Eulerian numbers for calculating compositional inverse of h(x) = (d/dx)^(-1) 1/g(x); iterated derivatives as infinitesimal generators of flows. Coefficients of a normalized Schwarzian derivative generating the Neretin polynomials: S(f) = (x^2/6) { D^2 log(f(x)) - (1/2) [D log(f(x))]^2 }. The Eta triangle. The Zeta triangle. The Beta triangle read by rows. The Lambda triangle The RSEG2 triangle. The EG1 triangle. The pg(n) sequence that is associated with the Eta triangle A160464. Numerators of the BG1[ -5,n] coefficients of the BG1 matrix Numerators of the column sums of the ZG1 matrix Numerators of the column sums of the LG1 matrix Decimal expansion of the higher-order exponential integral E(x, m=2, n=1) at x=1. Decimal expansion of Van der Pauw’s constant = Pi/log(2). Expansion of (1+x)*(3*x+1)/(1+x+x^2). Let CK(m) denote the complete elliptic integral of the first kind. a(n) is the n-th smallest integer k such that [(CK(1/k)] = [CK(1/(k-1)] + 1. Numbers m such that m and m+22 have the same sum of divisors. n such that the Moebius function take successively, from n, the values -1,0,-1,0,-1,0. Smallest m such that the Moebius function takes successively, from m, n values 1,0,1,0,... ending with 1 or 0. sigma(phi(n)) = rad(n), where phi(n)= Euler totient function, sigma(n)= sum of divisors of n, rad(n)= product of primes that divide n. Denominators of coefficients of a series, called f, related to Airy functions. Denominators of coefficients of a series, called g, related to Airy functions. Basic Multinomial Coefficients Triangle T(n,m) of the coefficients JacobiDC(x,y) = sum_{n>=0} sum_{m=0..n} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!. Triangle T(n,m) of the coefficients JacobiNC(x,y) = sum_{n>0} sum_{m=0..n-1} (-1)^m* T(n,m) *x^(2*n) *y^(2*m)/(2*n)!. a(n) = Sum_{k=0..n-1} cos(Pi*k/2)*binomial(n-1,k)*a(n-1-k)*a(k) for n > 0, a(0) = 1. Denominator of the rationals obtained from the e.g.f. D(1,x), a Debye function. Numerators of rationals with e.g.f. D(3,x), a Debye function. Numerators of rationals with e.g.f. D(4,x), a Debye function. Partition array in Abramowitz-Stegun order: Schur functions evaluated at 1. Partition array for the products of the hook lengths of Ferrers (Young) diagrams corresponding to the partitions of n, written in Abramowitz-Stegun order. Coefficients in the expansion Product_{ n>=1 } (1-q^(2n-1))^2/(1-q^(2n))^2. Irregular triangle which lists in row n the divisors of 2*n+1. Triangle entry T(n, m) gives the m-th contribution T(n, m)*sin((2*m+1)*v) to the coefficient of q^n in the Fourier expansion of Jacobi’s elliptic sn(u|k) function when expressed in the variables v = u/(2*K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions. Irregular triangle read by rows in which row n lists the divisors d of 2*n+1 (A274658), given the sign (-1)^(n + (d-1)/2). Triangle read by rows: T(n, m) gives the m-th contribution T(n, m)*cos((2*m+1)*v) to the coefficient of q^n in the Fourier expansion of Jacobi’s elliptic cn(u|k) function when expressed in the variables v = u/(2*K(k)/Pi) and q, the Jacobi nome, written as series in (k/4)^2. K is the real quarter period of elliptic functions. Numerators of partial sums of a hypergeometric series with value Pi/(sqrt(2)*(Gamma(5/8)*Gamma(7/8))^2) = A278144. Triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x^2 + 2*x + 1)^n + (x^2 - 1)*(x + 1)^n. Decimal expansion of Psi(3). Triangle read by rows, T(n, k) for 0 <= k <= n. T(n, 0) = 0^n; T(n, n) = n!; otherwise T(n, k) = (n + 1 - k)*(k - 1)!. Decimal expansion of Integral_{x=0..1} x/(exp(x)-1) dx.