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

0,3

Comments

Hankel transform is 3^n.

Links

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

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

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 *)

Prog

(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
(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
(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
(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

Crossrefs

Sequence in context: A269956 A153512 A369358 * A002891 A089875 A035617

Adjacent sequences: A127325 A127326 A127327 * A127329 A127330 A127331

Keyword

nonn

Author

Philippe Deléham, Mar 29 2007

Extensions

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

Status

approved