A383149 -id:A383149 - cp's OEIS frontend

A383163 Expansion of e.g.f. log(1 - (exp(2*x) - 1)/2)^2 / 2.

0, 0, 1, 9, 75, 690, 7290, 88536, 1223628, 19019664, 328908720, 6268688448, 130615236576, 2954657491968, 72128519473920, 1890266313945600, 52937770062975744, 1577901064699594752, 49877742373556336640, 1666688195869095124992, 58704547943954039672832

Offset: 0

Seiichi Manyama, Apr 18 2025

nonn

Column k=2 of A383149.

Cf. A000254.

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

a(n) = Sum{k=2..n} 2^(n-k) * Stirling2(n,k) * |Stirling1(k,2)|.

a(n) ~ sqrt(Pi) * 2^(n + 1/2) * n^(n - 1/2) * log(n) / (exp(n) * log(3)^n). - Vaclav Kotesovec, Apr 18 2025

A383164 Expansion of e.g.f. -log(1 - (exp(2*x) - 1)/2)^3 / 6.

Original entry on oeis.org

0, 0, 0, 1, 18, 255, 3555, 52290, 831684, 14405580, 271688580, 5562400800, 123123764808, 2933953637472, 74953425290016, 2044855241694720, 59361121229581440, 1827578437315965696, 59494057195888597248, 2042194772007257103360, 73731225467600254686720

Offset: 0

Seiichi Manyama, Apr 18 2025

nonn

Column k=3 of A383149.

Cf. A000399, A383166.

a(n) = sum(k=3, n, 2^(n-k)*stirling(n, k, 2)*abs(stirling(k, 3, 1)));

a(n) = Sum{k=3..n} 2^(n-k) * Stirling2(n,k) * |Stirling1(k,3)|.

a(n) ~ sqrt(Pi) * 2^(n - 1/2) * n^(n - 1/2) * log(n)^2 / (exp(n) * log(3)^n). - Vaclav Kotesovec, Apr 18 2025

cp's OEIS Frontend

A383163 Expansion of e.g.f. log(1 - (exp(2*x) - 1)/2)^2 / 2.

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