A156886 a(n) = Sum_{k=0..n} C(n,k)*C(3*n+k,k).

1, 5, 43, 416, 4239, 44485, 475780, 5156548, 56437231, 622361423, 6904185523, 76964141600, 861408728964, 9673849095708, 108954068684616, 1230185577016156, 13920106205444335, 157814104889538739

Offset: 0

Author

Paul Barry, Feb 17 2009

Keywords

• easy
• nonn

Comments

a(n)=[x^n](1+5x+9x^2+7x^3+2x^4)^n. The coefficients (1,5,9,7,2) are the 5th row of A029635.

Links

• Robert Israel, Table of n, a(n) for n = 0..937
• P. Barry, A Note on a Family of Generalized Pascal Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.4.

Crossrefs

Cf. A001850, A114496, A029635, A156887.

Programs

Maple

A156886 := proc(n)
    add(binomial(n,k)*binomial(3*n+k,k), k = 0..n);
end proc:
seq(A156886(n), n = 0..20); # Peter Bala, Feb 11 2018

Mathematica

a[n_] := Sum[ Binomial[n, k] Binomial[3n + k, k], {k, 0, n}]; Array[a, 21, 0] (* Robert G. Wilson v, Feb 11 2018 *)

Formula

From Peter Bala, Feb 11 2018: (Start)

a(n) = Sum_{k = 0..n} (-1)^(n-k)*C(n,k)*C(3*n+k,n)*2^k.

a(n) = Sum_{k = 0..n} C(n,k)*C(3*n,k)*2^(n-k),

12*n*(3*n-1)*(3*n-2)*(238*n^2 - 663*n + 457)*a(n) = 2*(150416*n^5 - 644640*n^4 + 1020351*n^3 - 734334*n^2 + 237007*n - 26880)*a(n-1) - (3*n-3)*(3*n-4)*(3*n-5)*(238*n^2 - 187*n + 32)*a(n-2). (End)

a(n) = P_n(0,2*n,3) where P_n(a,b,x) is the n-th Jacobi polynomial with parameters a and b. - Robert Israel, Feb 11 2018

a(n) ~ sqrt(1/3 + 11/(12*sqrt(7))) * ((316 + 119*sqrt(7))/54)^n / sqrt(Pi*n). - Vaclav Kotesovec, Jan 09 2023