A192989 Expansion of e.g.f.: exp((1+x)^3 - 1).

1, 3, 15, 87, 585, 4383, 35919, 318195, 3015441, 30354075, 322626159, 3603292047, 42120047385, 513557128503, 6512375759535, 85673471945067, 1166675225150241, 16413589529042355, 238151194659626319, 3558129109803374535

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..561

Formula

a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k) * 3^k.

Conjecture: a(n) -3*a(n-1) +6*(-n+1)*a(n-2) -3*(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 12 2014

Remark: the above conjectured recurrence is true and can be easily obtained from the e.g.f. - Emanuele Munarini, Aug 31 2017

a(n) ~ 3^(n/3-1/2) * exp(-2*n/3 + 3^(1/3)*n^(2/3) + 3^(-1/3)*n^(1/3) - 2/3) * n^(2*n/3) * (1 + 23/(54*(n/3)^(1/3)) + 3149/(29160*(n/3)^(2/3))). - Vaclav Kotesovec, Jul 15 2014

Example

E.g.f.: A(x) = 1 + 3*x + 15*x^2/2! + 87*x^3/3! + 585*x^4/4! +...

Mathematica

CoefficientList[Series[E^((1+x)^3-1), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 15 2014 *)
Table[Sum[ (-1)^(n - k) Abs[StirlingS1[n, k]] 3^k BellB[k], {k, 0, n}], {n, 0, 20}] (* Emanuele Munarini, Aug 31 2017 *)

Prog

(PARI) {a(n)=if(n<0, 0, n!*polcoeff(exp((1+x)^3-1+x*O(x^n)), n))}
(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)*3^k)}
(Maxima)
a(n) := sum((-1)^(n-k)*abs(stirling1(n, k))*3^k*belln(k), k, 0, n);
makelist(a(n), n, 0, 12); /* Emanuele Munarini, Aug 31 2017 */
(Magma) [(&+[3^k*Bell(k)*StirlingFirst(n, k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 25 2019
(Sage) [sum((-1)^(n-k)*3^k*bell_number(k)*stirling_number1(n, k) for k in (0..n)) for n in (0..20)] # G. C. Greubel, Jul 25 2019
(GAP) List([0..20], n-> Sum([0..n], k-> (-1)^(n-k)*3^k*Bell(k)* Stirling1(n, k) )); # G. C. Greubel, Jul 25 2019

Crossrefs

Cf. A000110, A000898, A008275.

Sequence in context: A359797 A246538 A132371 * A316666 A192253 A368972

Adjacent sequences: A192986 A192987 A192988 * A192990 A192991 A192992

Keyword

nonn

Author

Paul D. Hanna, Jul 13 2011

Status

approved