A349598 E.g.f. satisfies: log(A(x)) = exp(x*A(x)^2) - 1.
1, 1, 6, 71, 1279, 31142, 958127, 35674921, 1560207964, 78410153193, 4453247964775, 282086867840252, 19718661737739301, 1507855981764016549, 125211854842018500134, 11220898483255456505555, 1079389691811367897870339, 110936313685240067472613726
Offset: 0
Keywords
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..343
Programs
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Mathematica
a[n_] := Sum[(2*n + 1)^(k - 1)*StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Nov 23 2021 *) -
PARI
a(n) = sum(k=0, n, (2*n+1)^(k-1)*stirling(n, k, 2));
Formula
a(n) = Sum_{k=0..n} (2*n+1)^(k-1) * Stirling2(n,k).
a(n) ~ s * n^(n-1) / (2 * sqrt(1 + r*s^2) * exp(n) * r^n), where r = 0.1513832219344136560178112221696108323993292386502... and s = 1.52429184135463908701026733917578550814344591549... are roots of the system of equations (1 + log(s))*2*r*s^2 = 1, 2*r*s^2*exp(r*s^2) = 1. - Vaclav Kotesovec, Nov 25 2021
Equivalently, a(n) ~ n^(n-1) / (2*sqrt(1 + LambertW(1/2)) * LambertW(1/2)^n * exp(3*n + 1 - (n + 1/2)/LambertW(1/2))). - Vaclav Kotesovec, Nov 26 2021