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A202825 Expansion of e.g.f.: exp( (1+x)^5 - 1 ). 3
1, 5, 45, 485, 6145, 88245, 1403725, 24383525, 457473825, 9191615525, 196455592525, 4442277025125, 105787516038625, 2642880807687125, 69040011233566125, 1880443426122681125, 53268012941536530625, 1565875625728027213125, 47673392561258073158125 (list; graph; refs; listen; history; text; internal format)
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

Cf. A000110, A000898, A008275, A192989, A202824.

Sequence in context: A220877 A130976 A191095 * A195188 A232730 A151831

Adjacent sequences:  A202822 A202823 A202824 * A202826 A202827 A202828

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 25 2011

STATUS

approved

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Last modified May 11 10:21 EDT 2021. Contains 343788 sequences. (Running on oeis4.)