A127328 Inverse binomial transform of A026641; binomial transform of A127361.

1, 0, 3, 3, 15, 30, 99, 252, 747, 2064, 5973, 16995, 49089, 141414, 409755, 1188243, 3455811, 10064952, 29368377, 85809681, 251067645, 735446106, 2156695533, 6330729438, 18600079221, 54693760680, 160951905819, 473984678037, 1396755865527, 4118553190254

Offset: 0

Author

Philippe Deléham, Mar 29 2007

Keywords

• nonn

Comments

Hankel transform is 3^n.

Links

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

Programs

GAP

List([0..30], n-> Sum([0..n], k-> Sum([0..k], j-> (-1)^(n+j)* Binomial(k+j, j)*Binomial(n,k)))); # G. C. Greubel, Apr 30 2019

Magma

[ (&+[ (&+[(-1)^(n+j)*Binomial(k+j, j)*Binomial(n, k): j in [0..k]]): k in [0..n]]) : n in [0..30]]; // G. C. Greubel, Apr 30 2019

Mathematica

a[n_]:= Sum[(-1)^n*Sum[(-1)^j*Binomial[k+j, j], {j,0,k}]*Binomial[n, k], {k, 0, n}]; Table[a[n], {n, 0, 30}] (* G. C. Greubel, Apr 30 2019 *)

PARI

{a(n) = sum(k=0,n, sum(j=0,k, (-1)^(n+j)*binomial(k+j, j)* binomial(n, k)))}; \\ G. C. Greubel, Apr 30 2019

Sage

[sum(sum((-1)^(n+j)*binomial(k+j, j)*binomial(n, k) for j in (0..k)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Apr 30 2019

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..k} (-1)^(n+j)*binomial(k+j, j)*binomial(n, k). - G. C. Greubel, Apr 30 2019

a(n) ~ 3^n / sqrt(3*Pi*n). - Vaclav Kotesovec, Jul 20 2019

Extensions

Terms a(10) onward added by G. C. Greubel, Apr 30 2019