A202825 Expansion of e.g.f.: exp( (1+x)^5 - 1 ).
1, 5, 45, 485, 6145, 88245, 1403725, 24383525, 457473825, 9191615525, 196455592525, 4442277025125, 105787516038625, 2642880807687125, 69040011233566125, 1880443426122681125, 53268012941536530625, 1565875625728027213125, 47673392561258073158125
Offset: 0
Keywords
Examples
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).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..467
Programs
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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
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Magma
[(&+[5^k*Bell(k)*StirlingFirst(n,k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jul 25 2019
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Mathematica
Table[Sum[StirlingS1[n, k] 5^k BellB[k], {k, 0, n}], {n, 0, 20}] (* Emanuele Munarini, Sep 06 2017 *) -
Maxima
makelist(sum(stirling1(n,k)*5^k*belln(k),k,0,n),n,0,12); /* Emanuele Munarini, Sep 06 2017 */
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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)} -
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
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