OFFSET
0,2
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..467
FORMULA
a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k) * 5^k.
a(n+5) - 5*a(n+4) - 20*(n+4)*a(n+3) - 30*(n+3)*(n+4)*a(n+2) - 20*(n+2)*(n+3)*(n+4)* a(n+1) - 5*(n+1)*(n+2)*(n+3)*(n+4)*a(n) = 0. - Emanuele Munarini, Sep 06 2017
EXAMPLE
E.g.f.: A(x) = 1 + 5*x + 45*x^2/2! + 485*x^3/3! + 6145*x^4/4! +...
where A(x) = exp(5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5).
MATHEMATICA
Table[Sum[StirlingS1[n, k] 5^k BellB[k], {k, 0, n}], {n, 0, 20}] (* Emanuele Munarini, Sep 06 2017 *)
PROG
(PARI) {a(n)=n!*polcoeff(exp((1+x +x*O(x^n))^5-1), 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) * 5^k)}
(Maxima) makelist(sum(stirling1(n, k)*5^k*belln(k), k, 0, n), n, 0, 12); /* Emanuele Munarini, Sep 06 2017 */
(Magma) [(&+[5^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)*5^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)*5^k*Bell(k)* Stirling1(n, k) )); # G. C. Greubel, Jul 25 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 25 2011
STATUS
approved