A136658 Row sums of unsigned triangle A136656 and also of triangle 2*A136657.

1, 2, 10, 68, 580, 5912, 69784, 933200, 13912336, 228390560, 4088594464, 79186453568, 1648396356160, 36678170613632, 868239454798720, 21776352497954048, 576629116655862016, 16069766602389885440, 470015788927133039104, 14392014594072635786240

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..432

Formula

a(n) = Sum_{k=0..n} (-1)^n*A136656(n,k), n>=0.

E.g.f.: exp(x*(2-x)/(1-x)^2) (from Jabotinsky type triangle).

a(n) = Sum_{k=0..n} Stirling1(n, k) * Bell(k) * (-1)^(n-k) * 2^k. - Paul D. Hanna, Dec 25 2011

a(n) = (3*n-1)*a(n-1) - 3*(n-2)*(n-1)*a(n-2) + (n-3)*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Sep 25 2013

a(n) ~ 2^(1/6)*n^(n-1/6) * exp((n/2)^(1/3)+3*(n/2)^(2/3)-n-2/3) / sqrt(3) * (1 + 7/(27*(n/2)^(1/3)) - 422/(3645*(n/2)^(2/3))). - Vaclav Kotesovec, Sep 25 2013

Representation as special values of hypergeometric functions 2F2, in Maple notation: a(n) = (n+1)!*hypergeom([(1/2)*n+1, (1/2)*n+3/2], [3/2, 2], 1)*exp(-1), n = 1,2,... . - Karol A. Penson, Jul 28 2018

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*(j+1)!*a(n-j), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 01 2017

Mathematica

a[n_] := Sum[ StirlingS1[n, k] * BellB[k] * (-1)^(n-k) * 2^k, {k, 0, n}]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jul 09 2013, after Paul D. Hanna *)
Table[Sum[BellY[n, k, (Range[n] + 1)!], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Prog

(PARI)
{Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
{Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
{a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * (-1)^(n-k)*2^k)}
/* Paul D. Hanna, Dec 25 2011 */

Crossrefs

Cf. A049376, A136656, A136657.

Sequence in context: A361769 A082580 A231492 * A329475 A165968 A104098

Adjacent sequences: A136655 A136656 A136657 * A136659 A136660 A136661

Keyword

nonn,easy

Author

Wolfdieter Lang, Feb 22 2008

Status

approved