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
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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
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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}]
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;
A354123
Expansion of e.g.f. 1/(1 + log(1 - x))^4.
Original entry on oeis.org
1, 4, 24, 188, 1804, 20416, 265640, 3901320, 63776280, 1147796160, 22540858080, 479500074720, 10980929163360, 269298981833280, 7040446188020160, 195439047629422080, 5740498087530831360, 177855276360034736640, 5796391124741936993280
Offset: 0
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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)!*abs(stirling(n, k, 1)))/6;
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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