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A331725
E.g.f.: exp(x/(1 - x)) / (1 + x).
2
1, 0, 3, 4, 57, 216, 2755, 18348, 247569, 2368432, 35256771, 436248660, 7235178313, 108919083144, 2010150360387, 35421547781116, 723689454172065, 14543895730321248, 326843345169621379, 7354350135365751972, 180610925178770615001, 4488323611011676811320
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * k! * A000262(n-k).
a(n) ~ n^(n - 1/4) / (2^(3/2) * exp(1/2 - 2*sqrt(n) + n)). - Vaclav Kotesovec, Jan 26 2020
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
a(n) = Sum_{k=0..n} (-1)^k * A206703(n,k). - Alois P. Heinz, Feb 19 2022
a(n) = (-1)^n*n!*(1 - Sum_{j=0..n-1}((-1)^j*LaguerreL(j, 1, -1)/(j + 1))). - Peter Luschny, Feb 20 2022
MAPLE
A331725 := proc(n)
add((-1)^k*binomial(n, k)*k!*A000262(n-k), k=0..n) ;
end proc:
seq(A331725(n), n=0..42) ; # R. J. Mathar, Aug 20 2021
MATHEMATICA
nmax = 21; CoefficientList[Series[Exp[x/(1 - x)]/(1 + x), {x, 0, nmax}], x] Range[0, nmax]!
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}]
a[n_] := (-1)^n n! (1 - Sum[(-1)^j*LaguerreL[j, 1, -1]/(j+1), {j, 0, n-1}]);
Table[a[n], {n, 0, 21}] (* Peter Luschny, Feb 20 2022 *)
PROG
(PARI) seq(n)={Vec(serlaplace(exp(x/(1 - x) + O(x*x^n)) / (1 + x)))} \\ Andrew Howroyd, Jan 25 2020
(SageMath)
def gen_a():
F, L, S, N = 1, 1, 1, 1
while True:
yield F * S
L = gen_laguerre(N - 1, 1, -1) / N
S += L if F < 0 else -L
F *= -N; N += 1
a = gen_a(); print([next(a) for _ in range(21)]) # Peter Luschny, Feb 20 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Jan 25 2020
STATUS
approved