A034942 Binomial transform of A002054.

1, 6, 32, 163, 813, 4013, 19703, 96477, 471811, 2306316, 11274066, 55128021, 269688723, 1320047550, 6465047880, 31682288305, 155354522955, 762235643450, 3742013092520, 18380748447345, 90334417334095, 444186621323945, 2185193308168959, 10755153057295603, 52958560495897573

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• László Németh, Tetrahedron trinomial coefficient transform, arXiv:1905.13475 [math.CO], 2019.

Programs

Mathematica

Table[Sum[Binomial[n,k]*Binomial[2*k+3,k],{k,0,n}],{n,0,25}] (* Vaclav Kotesovec, Oct 08 2012 *)

PARI

a(n)=sum(k=0,n,binomial(n,k)*binomial(2*k+3,k)); \\ Joerg Arndt, May 04 2013

Formula

Recurrence: (n+3)*(3*n+1)*a(n) = 3*(6*n^2+15*n+11)*a(n-1) - 5*(n-1)*(3*n+4)*a(n-2). - Vaclav Kotesovec, Oct 08 2012

a(n) ~ 4*5^(n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Oct 08 2012