A349656 - OEIS

%I #26 Feb 16 2025 08:34:02

%S 1,1,-4,35,-515,10662,-284105,9255185,-356346618,15831168657,

%T -797090201295,44853942667096,-2789671436309939,190023794141566309,

%U -14069208182313480292,1124994237749880216439,-96618656489949875115879,8870165918232448251272870

%N E.g.f. satisfies: A(x)^2 * log(A(x)) = 1 - exp(-x).

%H Seiichi Manyama, <a href="/A349656/b349656.txt">Table of n, a(n) for n = 0..347</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F a(n) = (-1)^(n-1) * Sum_{k=0..n} (2*k-1)^(k-1) * Stirling2(n,k).

%F E.g.f.: A(x) = exp( LambertW(2*(1 - exp(-x)))/2 ).

%F G.f.: Sum_{k>=0} (-2*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).

%F a(n) ~ -(-1)^n * sqrt(2*exp(1) + 1) * sqrt(-log(2) + log(2 + exp(-1))) * n^(n-1) / (2 * exp(n + 1/2) * (-log(2) + log(2*exp(1) + 1) - 1)^n). - _Vaclav Kotesovec_, Nov 24 2021

%t a[n_] := (-1)^(n - 1) * Sum[(2*k - 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 27 2021 *)

%o (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, (2*k-1)^(k-1)*stirling(n, k, 2));

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(2*(1-exp(-x)))/2)))

%o (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (-2*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))

%Y Cf. A349650, A349652, A349654.

%Y Cf. A008277, A349585, A349657.

%K sign,changed

%O 0,3

%A _Seiichi Manyama_, Nov 23 2021