A354120
Expansion of e.g.f. 1/(1 - log(1 + x))^3.
Original entry on oeis.org
1, 3, 9, 30, 114, 492, 2388, 12912, 77016, 503112, 3570552, 27399600, 225729360, 1991996640, 18690559200, 186620451840, 1963991600640, 21914748541440, 255336518292480, 3155705206364160, 40209018105116160, 547746803311864320, 7525926332189130240
Offset: 0
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Table[Sum[(k+2)! * StirlingS1[n,k], {k,0,n}]/2, {n,0,35}] (* Vaclav Kotesovec, Jun 04 2022 *)
With[{nn=30},CoefficientList[Series[1/(1-Log[1+x])^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 16 2025 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^3))
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a(n) = sum(k=0, n, (k+2)!*stirling(n, k, 1))/2;
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
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Table[Sum[(k+3)! * StirlingS1[n,k], {k,0,n}]/6, {n,0,20}] (* Vaclav Kotesovec, Jun 04 2022 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+x))^4))
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a(n) = sum(k=0, n, (k+3)!*stirling(n, k, 1))/6;
A320344
Expansion of e.g.f. log(1 + x)/(1 - log(1 + x))^2.
Original entry on oeis.org
0, 1, 3, 8, 26, 94, 406, 1896, 10440, 59472, 405264, 2673648, 22396128, 160828368, 1704287568, 11993279232, 177349981824, 957018589056, 25766036316288, 33555346603776, 5403108443855616, -28811285794990080, 1643455634670489600, -21001090458387594240, 692074413969784289280
Offset: 0
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seq(n!*coeff(series(log(1+x)/(1-log(1+x))^2,x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 29 2019
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nmax = 24; CoefficientList[Series[Log[1 + x]/(1 - Log[1 + x])^2, {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k k!, {k, 0, n}], {n, 0, 24}]
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my(N=40, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N, k*log(1+x)^k)))) \\ Seiichi Manyama, Apr 22 2022
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
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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}]
Showing 1-4 of 4 results.
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