Noe, Tony D. On the divisibility of generalized central trinomial coefficients. (English) Zbl 1102.05009 J. Integer Seq. 9, No. 2, Article 06.2.7, 12 p. (2006). 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)]. Cited in 1 ReviewCited in 17 Documents MSC: 05A19 Combinatorial identities, bijective combinatorics 11A07 Congruences; primitive roots; residue systems 05A10 Factorials, binomial coefficients, combinatorial functions Keywords:Schur and Holt congruences; Legendre polynomials Citations:Zbl 1163.11310 Software:OEIS PDFBibTeX XMLCite \textit{T. D. Noe}, J. Integer Seq. 9, No. 2, Article 06.2.7, 12 p. (2006; Zbl 1102.05009) 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).