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%I #18 Feb 20 2022 11:09:10
%S 1,0,3,4,57,216,2755,18348,247569,2368432,35256771,436248660,
%T 7235178313,108919083144,2010150360387,35421547781116,723689454172065,
%U 14543895730321248,326843345169621379,7354350135365751972,180610925178770615001,4488323611011676811320
%N E.g.f.: exp(x/(1 - x)) / (1 + x).
%H Alois P. Heinz, <a href="/A331725/b331725.txt">Table of n, a(n) for n = 0..444</a>
%F a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * k! * A000262(n-k).
%F a(n) ~ n^(n - 1/4) / (2^(3/2) * exp(1/2 - 2*sqrt(n) + n)). - _Vaclav Kotesovec_, Jan 26 2020
%F D-finite with recurrence a(n) +(-n+1)*a(n-1) -(n-1)*(n+1)*a(n-2) +(n-1)*(n-2)^2*a(n-3)=0. - _R. J. Mathar_, Aug 20 2021
%F a(n) = Sum_{k=0..n} (-1)^k * A206703(n,k). - _Alois P. Heinz_, Feb 19 2022
%F a(n) = (-1)^n*n!*(1 - Sum_{j=0..n-1}((-1)^j*LaguerreL(j, 1, -1)/(j + 1))). - _Peter Luschny_, Feb 20 2022
%p A331725 := proc(n)
%p add((-1)^k*binomial(n,k)*k!*A000262(n-k),k=0..n) ;
%p end proc:
%p seq(A331725(n),n=0..42) ; # _R. J. Mathar_, Aug 20 2021
%t nmax = 21; CoefficientList[Series[Exp[x/(1 - x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
%t A000262[n_] := If[n == 0, 1, n! Sum[Binomial[n - 1, k]/(k + 1)!, {k, 0, n - 1}]]; a[n_] := Sum[(-1)^k Binomial[n, k] k! A000262[n - k], {k, 0, n}]; Table[a[n], {n, 0, 21}]
%t a[n_] := (-1)^n n! (1 - Sum[(-1)^j*LaguerreL[j, 1, -1]/(j+1), {j,0,n-1}]);
%t Table[a[n], {n, 0, 21}] (* _Peter Luschny_, Feb 20 2022 *)
%o (PARI) seq(n)={Vec(serlaplace(exp(x/(1 - x) + O(x*x^n)) / (1 + x)))} \\ _Andrew Howroyd_, Jan 25 2020
%o (SageMath)
%o def gen_a():
%o F, L, S, N = 1, 1, 1, 1
%o while True:
%o yield F * S
%o L = gen_laguerre(N - 1, 1, -1) / N
%o S += L if F < 0 else -L
%o F *= -N; N += 1
%o a = gen_a(); print([next(a) for _ in range(21)]) # _Peter Luschny_, Feb 20 2022
%Y Cf. A000262, A002720, A009940, A133942, A182386, A206703.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 25 2020
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