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On the divisibility of generalized central trinomial coefficients. (English) Zbl 1102.05009

Summary: We present several methods of computing sequences of generalized central trinomial coefficients. We generalize the Schur and Holt congruences for Legendre polynomials in order to prove divisibility properties of these sequences and a conjecture of E. Deutsch and B. E. Sagan [J. Number Theory 117, No. 1, 191–215 (2006; Zbl 1163.11310)].

MSC:

05A19 Combinatorial identities, bijective combinatorics
11A07 Congruences; primitive roots; residue systems
05A10 Factorials, binomial coefficients, combinatorial functions

Citations:

Zbl 1163.11310

Software:

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Full Text: EuDML EMIS

Online Encyclopedia of Integer Sequences:

Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2.
Numerators in expansion of 1/sqrt(1-x).
Central Delannoy numbers: a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k).
Central trinomial coefficients: largest coefficient of (1 + x + x^2)^n.
n*a(n) = 2*(2*n-1)*a(n-1) + 4*(n-1)*a(n-2) with a(0) = 1.
Expansion of 1/sqrt(1 - 10*x + x^2).
Expansion of 1/sqrt(1 - 4*x + 16*x^2).
a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k).
a(n) = 2^n * (2*n)! / (n!)^2.
Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).
Binomial transform of central Delannoy numbers A001850.
Expansion of e.g.f. exp(4x) * I_0(2x).
Coefficients of 1/(1-2x-7x^2)^(1/2); also, a(n) is the central coefficient of (1+x+2x^2)^n.
Coefficients of 1/sqrt(1 - 2*x - 11*x^2); also, a(n) is the central coefficient of (1 + x + 3*x^2)^n.
G.f.: 1/(1-2x-15x^2)^(1/2); also, a(n) is the central coefficient of (1+x+4x^2)^n.
Coefficients of 1/sqrt(1-4*x-8*x^2); also, a(n) is the central coefficient of (1+2*x+3*x^2)^n.
P_n(7), where P_n is n-th Legendre polynomial; also, a(n) = central coefficient of (1 + 7*x + 12*x^2)^n.
P_n(9), where P_n is n-th Legendre polynomial; also, a(n) = central coefficient of (1 + 9*x + 20*x^2)^n.
Coefficients of 1/(1-4x-16x^2)^(1/2); also, a(n) is the central coefficient of (1+2x+5x^2)^n.
Coefficients of expansion of 1/sqrt(1 - 10*x + 9*x^2); also, a(n) is the central coefficient of (1 + 5*x + 4*x^2)^n.
G.f.: 1/(1-2x-19x^2)^(1/2).
G.f. : 1/(1-2x-23x^2)^(1/2).
Expansion of 1/(1-2x-31x^2)^(1/2).
Expansion of 1/sqrt(1 - 2*x + 5*x^2).
Expansion of 1/sqrt(1 - 2*x + 9*x^2).
Expansion of 1/sqrt(1 - 2x + 13x^2).
Expansion of 1/sqrt(1-2x+17x^2).
Expansion of 1/sqrt(1-4x+8x^2).
Expansion of 1/sqrt(1 - 4*x + 12*x^2).
Expansion of 1/sqrt(1-4x+20x^2).
Expansion of 1/sqrt(1-6x+13x^2).
Expansion of 1/sqrt(1 - 6x + 17x^2).
Expansion of 1/sqrt(1 - 6x + 21x^2).
Expansion of 1/sqrt(1 - 6*x + 25*x^2).
Expansion of 1/(sqrt(1-3*x)*sqrt(1-7*x)).
Expansion of 1/(sqrt(1-4*x)*sqrt(1-8*x)).
a(n) = 4^n*(2*n)!/(n!)^2.
Expansion of 1/sqrt(1-2x-47x^2).
Expansion of 1/sqrt(1-2x-59x^2).
Expansion of 1/sqrt(1 - 2*x - 63*x^2).
Expansion of 1/sqrt(1-2x-95x^2).
Expansion of 1/sqrt(1-8x-4x^2).
Expansion of 1/sqrt(1-6x-11x^2).
Expansion of 1/sqrt(1 - 4*x - 12*x^2).
a(n) = 3^n*(2*n)!/(n!)^2.
Expansion of 1/(sqrt(1-5x)*sqrt(1-9x)).
Expansion of 1/sqrt(1-8x-8x^2).
Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial transform. Same as interpolating bilateral Schroeder paths (A026375) with the central binomial coefficients (A000984).
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1-4*k)*x^2).
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/sqrt(1 - 2*x + (1+4*k)*x^2).