A383172 -id:A383172 - cp's OEIS frontend

A383170 Expansion of e.g.f. -log(1 + log(1 - 2*x)/2).

0, 1, 3, 16, 122, 1208, 14704, 212336, 3547984, 67337728, 1430990976, 33664165632, 868592478720, 24390846882816, 740570519159808, 24177326011834368, 844599686386919424, 31438092340685144064, 1242230898248798896128, 51933512200489564962816, 2290351520336982559358976

Offset: 0

Seiichi Manyama, Apr 18 2025

nonn

Cf. A383171, A383172.

Cf. A003713, A227917.

a(n) = sum(k=1, n, 2^(n-k)*abs(stirling(n, k, 1)*stirling(k, 1, 1)));

a(n) = Sum_{k=1..n} 2^(n-k) * |Stirling1(n,k) * Stirling1(k,1)|.

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/2) / (exp(1) - exp(-1))^n. - Vaclav Kotesovec, Apr 18 2025

A383171 Expansion of e.g.f. log(1 + log(1 - 2*x)/2)^2 / 2.

Original entry on oeis.org

0, 0, 1, 9, 91, 1090, 15298, 247352, 4537132, 93195696, 2120623984, 52973194560, 1441635171040, 42464913775232, 1346297567292416, 45715740985471744, 1655552663185480448, 63698261991541393408, 2595107348458704209920, 111613055867327344582656

Offset: 0

Seiichi Manyama, Apr 18 2025

nonn

Cf. A383170, A383172.

Cf. A341587, A383163.

a(n) = sum(k=2, n, 2^(n-k)*abs(stirling(n, k, 1)*stirling(k, 2, 1)));

a(n) = Sum_{k=2..n} 2^(n-k) * |Stirling1(n,k) * Stirling1(k,2)|.

a(n) ~ sqrt(Pi) * log(n) * 2^(n + 1/2) * n^(n - 1/2) / (exp(1) - exp(-1))^n * (1 + (gamma + log(2) - log(exp(2)-1))/log(n)), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Apr 18 2025

cp's OEIS Frontend

A383170 Expansion of e.g.f. -log(1 + log(1 - 2*x)/2).

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