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%I #48 Dec 01 2021 09:25:32
%S 1,0,-1,-1,2,9,9,-50,-267,-413,2180,17731,50533,-110176,-1966797,
%T -9938669,-8638718,278475061,2540956509,9816860358,-27172288399,
%U -725503033401,-5592543175252,-15823587507881,168392610536153,2848115497132448,20819319685262839
%N E.g.f.: exp(1 + x - exp(x)).
%H Seiichi Manyama, <a href="/A293037/b293037.txt">Table of n, a(n) for n = 0..593</a>
%F a(n) = exp(1) * Sum_{k>=0} (-1)^k*(k + 1)^n/k!. - _Ilya Gutkovskiy_, Jun 13 2019
%F a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -1). - _Vaclav Kotesovec_, Jul 06 2020
%F a(0) = 1; a(n) = - Sum_{k=0..n-2} binomial(n-1,k) * a(k). - _Seiichi Manyama_, Aug 02 2021
%p f:= series(exp(1 + x - exp(x)), x= 0, 101): seq(factorial(n) * coeff(f, x, n), n = 0..30); # _Muniru A Asiru_, Oct 31 2017
%p # second Maple program:
%p b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
%p add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
%p end:
%p a:= n-> b(n+1, 1):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Dec 01 2021
%t m = 26; Range[0, m]! * CoefficientList[Series[Exp[1 + x - Exp[x]], {x, 0, m}], x] (* _Amiram Eldar_, Jul 06 2020 *)
%t Table[Sum[Binomial[n, k] * BellB[k, -1], {k, 0, n}], {n, 0, 30}] (* _Vaclav Kotesovec_, Jul 06 2020 *)
%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-exp(x)+1+x)))
%o (PARI) a(n) = if(n==0, 1, -sum(k=0, n-2, binomial(n-1, k)*a(k))); \\ _Seiichi Manyama_, Aug 02 2021
%Y Column k=1 of A293051.
%Y Column k=1 of A335977.
%Y Cf. A000587 (k=0), this sequence (k=1), A293038 (k=2), A293039 (k=3), A293040 (k=4).
%Y Cf. A000296, A014182, A193683, A193684, A196835.
%K sign
%O 0,5
%A _Seiichi Manyama_, Sep 28 2017
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