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A202825 Expansion of e.g.f.: exp( (1+x)^5 - 1 ).

%I A202825 #18 Sep 08 2022 08:46:01
%S A202825 1,5,45,485,6145,88245,1403725,24383525,457473825,9191615525,
%T A202825 196455592525,4442277025125,105787516038625,2642880807687125,
%U A202825 69040011233566125,1880443426122681125,53268012941536530625,1565875625728027213125,47673392561258073158125
%N A202825 Expansion of e.g.f.: exp( (1+x)^5 - 1 ).
%H A202825 Seiichi Manyama, <a href="/A202825/b202825.txt">Table of n, a(n) for n = 0..467</a>
%F A202825 a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(k) * 5^k.
%F A202825 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
%e A202825 E.g.f.: A(x) = 1 + 5*x + 45*x^2/2! + 485*x^3/3! + 6145*x^4/4! +...
%e A202825 where A(x) = exp(5*x + 10*x^2 + 10*x^3 + 5*x^4 + x^5).
%t A202825 Table[Sum[StirlingS1[n, k] 5^k BellB[k], {k, 0, n}], {n, 0, 20}] (* _Emanuele Munarini_, Sep 06 2017 *)
%o A202825 (PARI) {a(n)=n!*polcoeff(exp((1+x +x*O(x^n))^5-1),n)}
%o A202825 (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}
%o A202825 {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}
%o A202825 {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(k) * 5^k)}
%o A202825 (Maxima) makelist(sum(stirling1(n,k)*5^k*belln(k),k,0,n),n,0,12); /* _Emanuele Munarini_, Sep 06 2017 */
%o A202825 (Magma) [(&+[5^k*Bell(k)*StirlingFirst(n,k): k in [0..n]]): n in [0..20]]; // _G. C. Greubel_, Jul 25 2019
%o A202825 (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
%o A202825 (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
%Y A202825 Cf. A000110, A000898, A008275, A192989, A202824.
%K A202825 nonn
%O A202825 0,2
%A A202825 _Paul D. Hanna_, Dec 25 2011