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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 13 2011
STATUS
approved