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A309085 a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!. 3

%I

%S 1,-4,12,-20,-20,172,108,-2388,-3220,47532,161900,-1062740,-8532628,

%T 13623212,431041132,1206169260,-17833021588,-169685043796,

%U 180187176044,13462762665132,79377664422252,-553096696140884,-11670986989785492,-44371854928405844,829755609457185644

%N a(n) = exp(4) * Sum_{k>=0} (-4)^k*k^n/k!.

%H Seiichi Manyama, <a href="/A309085/b309085.txt">Table of n, a(n) for n = 0..564</a>

%F G.f.: Sum_{j>=0} (-4)^j*x^j / Product_{k=1..j} (1 - k*x).

%F E.g.f.: exp(4*(1 - exp(x))).

%F a(n) = Sum_{k=0..n} (-4)^k * Stirling2(n,k).

%t Table[Exp[4] Sum[(-4)^k k^n/k!, {k, 0, Infinity}], {n, 0, 24}]

%t Table[BellB[n, -4], {n, 0, 24}]

%t nmax = 24; CoefficientList[Series[Sum[(-4)^j x^j/Product[(1 - k x), {k, 1, j}] , {j, 0, nmax}], {x, 0, nmax}], x]

%t nmax = 24; CoefficientList[Series[Exp[4 (1 - Exp[x])], {x, 0, nmax}], x] Range[0, nmax]!

%o (MAGMA) [1] cat [(&+[((-4)^k*StirlingSecond(m,k)):k in [0..m]]):m in [1..24]]; // _Marius A. Burtea_, Jul 11 2019

%o (PARI) a(n) = sum(k=0, n, (-4)^k * stirling(n,k,2)); \\ _Michel Marcus_, Jul 12 2019

%Y Column k = 4 of A292861.

%Y Cf. A000587, A078944, A213170, A309084, A318179.

%K sign

%O 0,2

%A _Ilya Gutkovskiy_, Jul 11 2019

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Last modified October 22 10:24 EDT 2019. Contains 328317 sequences. (Running on oeis4.)