A137220 a(n) = (A126086(n) + 3*A001850(n) + 2)/6.

1, 4, 75, 2712, 116681, 5366384, 256461703, 12582521536, 629390010177, 31955248465164, 1641724961412515, 85159811886281576, 4452782349821587705, 234393562420377364008, 12409423916987553634575, 660253088667255226947072

Offset: 0

Author

Vladeta Jovovic, Mar 06 2008, Mar 16 2008

Keywords

• easy
• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Column k=3 of A330942.

Cf. A047665, A137219.

Programs

Maple

A126086 := proc(n) local x,y,z ; coeftayl(coeftayl(coeftayl(1/(1-x-y-z-x*y-x*z-y*z-x*y*z),z=0,n),y=0,n),x=0,n) ; end: A001850 := proc(n) local k ; add(binomial(n,k)*binomial(n+k,k),k=0..n) ; end: A137220 := proc(n) (A126086(n)+3*A001850(n)+2)/6 ; end: seq(A137220(n),n=0..30) ; # R. J. Mathar, Apr 01 2008

Mathematica

T[n_, k_] := With[{m = n k}, Sum[Binomial[Binomial[j, n] + k - 1, k] Sum[ (-1)^(i - j) Binomial[i, j], {i, j, m}], {j, 0, m}]];
Table[T[n, 3], {n, 0, 15}] (* Jean-François Alcover, Apr 10 2020, after Andrew Howroyd *)

PARI

a(n) = {sum(j=0, 3*n, binomial(binomial(j,n)+2, 3) * sum(i=j, 3*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

Sage

@CachedFunction
def A137220(n): return round( -sum( binomial(-binomial(j, n), 3)/2^(j+1) for j in (0..500) ) )
[A137220(n) for n in (0..30)] # G. C. Greubel, Jan 05 2022

Formula

a(n) = -Sum_{m>=0} binomial(-binomial(m,n),3)/2^(m+1).

a(n) = A137219(n) + A001850(n). - R. J. Mathar, Apr 01 2008

a(n) = Sum_{j=0..3*n} binomial(binomial(j,n)+2, 3) * (Sum_{i=j..3*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

Extensions

More terms from R. J. Mathar, Apr 01 2008