A354120 -id:A354120 - cp's OEIS frontend

A354122 Expansion of e.g.f. 1/(1 + log(1 - x))^3.

1, 3, 15, 102, 870, 8892, 105708, 1431168, 21722136, 365105928, 6729341832, 134915992560, 2922576142320, 68013701197920, 1692075061072800, 44810389419079680, 1258472984174461440, 37357062009383877120, 1168635883239630120960, 38424619272539153157120

Offset: 0

Seiichi Manyama, May 17 2022

nonn

Cf. A007840, A052801, A354123.

Cf. A226515, A354120.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x))^3))

a(n) = sum(k=0, n, (k+2)!*abs(stirling(n, k, 1)))/2;

a(n) = (1/2) * Sum_{k=0..n} (k + 2)! * |Stirling1(n,k)|.
a(n) ~ sqrt(Pi/2) * n^(n + 5/2) / (exp(1) - 1)^(n+3). - Vaclav Kotesovec, Jun 04 2022
a(0) = 1; a(n) = Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

Original entry on oeis.org

1, 2, 4, 10, 30, 108, 444, 2112, 11040, 65712, 414816, 2992944, 21876816, 188936928, 1527813216, 15991733376, 133364903040, 1794144752640, 13329036288000, 270750383400960, 1167153128110080, 57074973648030720, -103080839984916480, 17319631144046423040, -171982551742151685120

Offset: 0

Ilya Gutkovskiy, Jul 25 2018

sign

Exponential self-convolution of A006252.

Seiichi Manyama, Table of n, a(n) for n = 0..451
N. J. A. Sloane, Transforms

Cf. A005444, A005649, A006252, A052801, A089064.

Cf. A006252, A354120, A354121.

a:=series(1/(1 - log(1 + x))^2, x=0, 25): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 26 2019

nmax = 24; CoefficientList[Series[1/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (k + 1)!, {k, 0, n}], {n, 0, 24}]

a(n) = Sum_{k=0..n} Stirling1(n,k)*(k + 1)!.
a(n) ~ n! * 2 * (-1)^(n+1) / (n * log(n)^3) * (1 - 3*(gamma+1) / log(n) + (6*gamma^2 + 12*gamma + 6 - Pi^2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, May 15 2022
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

Original entry on oeis.org

1, 4, 16, 68, 316, 1616, 9080, 55800, 373080, 2699520, 21035040, 175708320, 1566916320, 14862171840, 149429426880, 1587766126080, 17779538050560, 209295747832320, 2583920845209600, 33389139008678400, 450642388471395840, 6342869733912760320

Offset: 0

Seiichi Manyama, May 17 2022

sign

a(46) is negative. - Vaclav Kotesovec, Jun 04 2022

It appears that a(n) is negative for even n >= 46. - Felix Fröhlich, Jun 04 2022

Vaclav Kotesovec, Table of n, a(n) for n = 0..450

Cf. A006252, A317280, A354120.

Cf. A226738, A354123.

Table[Sum[(k+3)! * StirlingS1[n,k], {k,0,n}]/6, {n,0,20}] (* Vaclav Kotesovec, Jun 04 2022 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^4))

a(n) = sum(k=0, n, (k+3)!*stirling(n, k, 1))/6;

a(n) = (1/6) * Sum_{k=0..n} (k + 3)! * Stirling1(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (3 * k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A382840 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * Stirling1(n,k) * k!.

Original entry on oeis.org

1, 1, 4, 30, 316, 4290, 71268, 1400112, 31750416, 816215760, 23455342560, 745073660496, 25924233481056, 980518650296640, 40054724743501440, 1757539560656401920, 82439565962427760896, 4116529729771939393920, 218017561353648160158720, 12206586491422209675532800

Offset: 0

Ilya Gutkovskiy, Apr 06 2025

nonn

Cf. A006252, A305919, A308565, A317280, A354120, A354121, A382830.

Table[Sum[Binomial[n + k - 1, k] StirlingS1[n, k] k!, {k, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[1/(1 - Log[1 + x])^n, {x, 0, n}], {n, 0, 19}]

a(n) = n! * [x^n] 1 / (1 - log(1 + x))^n.

a(n) ~ n^n / (sqrt(1 + LambertW(1)) * 2^n * exp(n) * (cosh(LambertW(1)) - 1)^n). - Vaclav Kotesovec, Apr 07 2025

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A354122 Expansion of e.g.f. 1/(1 + log(1 - x))^3.

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A317280 Expansion of e.g.f. 1/(1 - log(1 + x))^2.

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A354121 Expansion of e.g.f. 1/(1 - log(1 + x))^4.

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A382840 a(n) = Sum_{k=0..n} binomial(n+k-1,k) * Stirling1(n,k) * k!.

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