A127905 Construct triangle in which n-th row is obtained by expanding (1+x+x^3)^n and take the next-to-central column.

0, 1, 2, 3, 8, 25, 66, 168, 456, 1269, 3490, 9581, 26544, 73944, 206220, 576045, 1613264, 4527661, 12725946, 35818135, 100950440, 284869263, 804726934, 2275500998, 6440230392, 18242735800, 51714552656

Offset: 0

Author

Paul Barry, Feb 05 2007

Keywords

• easy
• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A005717.

Programs

Magma

[0] cat [n*(&+[Binomial(n-1,3*k)*Binomial(3*k,k)/(2*k+1): k in [0..Floor((n-1)/3)]]): n in [1..30]]; // G. C. Greubel, Apr 30 2018

Maple

A127905 := proc(n)
    n*add(binomial(n-1,3*k)*binomial(3*k,k)/(2*k+1),k=0..floor((n-1)/3)) ;
end proc: # R. J. Mathar, Feb 23 2015

Mathematica

Table[n*Sum[Binomial[n-1,3*k]*Binomial[3*k,k]/(2*k+1), {k, 0, Floor[(n -1)/3]}], {n, 0, 50}] (* G. C. Greubel, Apr 30 2018 *)

PARI

a(n)=if(n<0, 0, polcoeff((1+x+x^3)^n, n-1));

PARI

a(n)=if(n<0, 0, n++; n*polcoeff(serreverse(x/(1+x+x^3)+x*O(x^n)), n))

Formula

a(n) = n*A071879(n-1).

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

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

a(n) = Sum_{k=0..n-1} Sum_{j=0..floor(k/3)} C(k,3*j)*C(3*j+1,j).

Conjecture: 2*(2*n+1)*(n-1)^2*a(n) -2*n*(6*n^2-12*n+5)*a(n-1) +6*n*(n-1)*(2*n-3)*a(n-2) -31*n*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Feb 23 2015

a(n) ~ (1 + 3/2^(2/3))^(n + 1/2) / sqrt(12*Pi*n). - Vaclav Kotesovec, May 01 2018

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009