A349585 E.g.f. satisfies: A(x) * log(A(x)) = 1 - exp(-x).
1, 1, -2, 8, -59, 642, -9112, 158839, -3279880, 78250188, -2117569181, 64082989720, -2144319848772, 78609355884893, -3133061858717806, 134884905211588892, -6238095343894356675, 308427209934965151158, -16234730389499986865092, 906409067599064528054343
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..378
- Eric Weisstein's World of Mathematics, Lambert W-Function.
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, (m-1)^(m-1), m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> (-1)^(n-1)*b(n, 0): seq(a(n), n=0..20); # Alois P. Heinz, Aug 03 2022 -
Mathematica
a[n_] := (-1)^(n - 1) * Sum[If[k == 1, 1, (k - 1)^(k - 1)]*StirlingS2[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Nov 23 2021 *) -
PARI
a(n) = (-1)^(n-1)*sum(k=0, n, (k-1)^(k-1)*stirling(n, k, 2));
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(1-exp(-x))))) -
PARI
my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))
Formula
a(n) = (-1)^(n-1) * Sum_{k=0..n} (k-1)^(k-1) * Stirling2(n,k).
E.g.f.: A(x) = exp( LambertW(1 - exp(-x)) ).
G.f.: Sum_{k>=0} (-k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
a(n) ~ -(-1)^n * sqrt(1 + exp(1)) * n^(n-1) / (exp(n+1) * (log(1 + exp(1)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Dec 05 2021