A000558 -id:A000558 - cp's OEIS frontend

A383207 Expansion of e.g.f. (exp(f(x)) - 1)^2 / 2, where f(x) = (exp(2*x) - 1)/2.

0, 0, 1, 9, 71, 575, 4957, 45829, 454015, 4804191, 54094749, 645720757, 8142419727, 108110708511, 1506969153757, 21993472779461, 335257957315199, 5325979566073919, 87999598425114045, 1509471498829147637, 26835040585117438415, 493677094649876461759, 9384926300821643459133

Offset: 0

Seiichi Manyama, Apr 19 2025

nonn

Column k=2 of A383206.

Cf. A000558.

a(n) = sum(k=2, n, 2^(n-k)*stirling(n, k, 2)*stirling(k, 2, 2));

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

A341586 E.g.f.: (exp(1 - exp(x)) - 1)^2 / 2.

Original entry on oeis.org

1, 0, -4, -5, 22, 98, -5, -1458, -5136, 9053, 161328, 549822, -1954067, -30099188, -114161728, 500200027, 8875931202, 42311243830, -149028931789, -3816065804086, -24704581255020, 33033659868037, 2184285021783940, 20047242475274290, 30117550563701293

Offset: 2

Ilya Gutkovskiy, Feb 15 2021

sign

Cf. A000225, A000558, A000587, A008277, A213170, A309775.

nmax = 26; CoefficientList[Series[(Exp[1 - Exp[x]] - 1)^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
Table[Sum[(-1)^k StirlingS2[n, k] StirlingS2[k, 2], {k, 2, n}], {n, 2, 26}]

a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * Stirling2(k, 2).
a(n) = Sum_{k=2..n} (-1)^k * Stirling2(n, k) * (2^(k-1) - 1).
a(n) = Sum_{k=1..n-1} binomial(n-1, k) * A000587(k) * A000587(n-k).

cp's OEIS Frontend

A383207 Expansion of e.g.f. (exp(f(x)) - 1)^2 / 2, where f(x) = (exp(2*x) - 1)/2.

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