A375952 -id:A375952 - cp's OEIS frontend

A375949 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(4/3).

1, 4, 32, 368, 5520, 102064, 2242832, 57095728, 1652211600, 53559908784, 1922581295632, 75700072208688, 3243905700776080, 150289130386531504, 7485459789379535632, 398857142195958963248, 22639650637589839298960, 1363772478150606703714224

Offset: 0

Seiichi Manyama, Sep 03 2024

nonn

Cf. A032033, A346982, A365558, A375952.

Cf. A005649, A007559, A375948.

nmax=17; CoefficientList[Series[1 / (4 - 3 * Exp[x])^(4/3),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)

a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k+1)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} A007559(k+1) * Stirling2(n,k).

a(n) ~ 3 * sqrt(Pi) * n^(n + 5/6) / (2^(13/6) * Gamma(1/3) * log(4/3)^(n + 4/3) * exp(n)). - Vaclav Kotesovec, Sep 06 2024

A375992 Expansion of e.g.f. (4 - 3 * exp(x))^(4/3).

Original entry on oeis.org

1, -4, 0, 16, 112, 976, 11760, 184656, 3566192, 81556176, 2152839920, 64389871696, 2151410517872, 79406805184976, 3208188040810480, 140812644820877136, 6671575179144279152, 339348322285418119376, 18443287953728909235440, 1066619199816333440144976

Offset: 0

Seiichi Manyama, Sep 05 2024

Cf. A032033, A346982, A365558, A375949, A375952, A375993.

Cf. A375988.

a(n) = sum(k=0, n, prod(j=0, k-1, 3*j-4)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j-4)) * Stirling2(n,k).

A375993 Expansion of e.g.f. (4 - 3 * exp(x))^(5/3).

Original entry on oeis.org

1, -5, 5, 35, 165, 1075, 10805, 152035, 2719365, 58547475, 1469512405, 42082036035, 1353220758565, 48264167285875, 1890433757030005, 80656857839376035, 3723074712045197765, 184851684577600696275, 9822823990059902723605, 556226222504163445932035

Offset: 0

Seiichi Manyama, Sep 05 2024

Cf. A032033, A346982, A365558, A375949, A375952, A375992.

a(n) = sum(k=0, n, prod(j=0, k-1, 3*j-5)*stirling(n, k, 2));

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j-5)) * Stirling2(n,k).

a(n) ~ 5 * sqrt(Pi) * 2^(29/6) * n^(n - 13/6) / (9 * Gamma(1/3) * exp(n) * log(4/3)^(n - 5/3)). - Vaclav Kotesovec, Sep 06 2024

cp's OEIS Frontend

A375949 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(4/3).

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A375992 Expansion of e.g.f. (4 - 3 * exp(x))^(4/3).

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A375993 Expansion of e.g.f. (4 - 3 * exp(x))^(5/3).

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