A382830
a(n) = Sum_{k=0..n} binomial(n+k-1,k) * |Stirling1(n,k)| * k!.
Original entry on oeis.org
1, 1, 8, 102, 1804, 40890, 1131108, 36948240, 1391945616, 59411849040, 2833582748160, 149347596487056, 8620256620495584, 540775669746661440, 36636074309252234880, 2665704585421541790720, 207329122282259073044736, 17165075378189396045777280, 1507206260097615729874083840
Offset: 0
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Table[Sum[Binomial[n + k - 1, k] Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[1/(1 + Log[1 - x])^n, {x, 0, n}], {n, 0, 18}]
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}]
A382847
a(n) = Sum_{k=0..n} binomial(n+k-1,k) * (Stirling2(n,k) * k!)^2.
Original entry on oeis.org
1, 1, 14, 579, 48044, 6647405, 1379024730, 400315753159, 154879704709784, 77018569697097009, 47863427797633958630, 36348262891572161261963, 33119479438137288670256964, 35660343372397246917403353013, 44791475616825872944740798413234, 64911462519379469821754507087299215
Offset: 0
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Table[Sum[Binomial[n + k - 1, k] (StirlingS2[n, k] k!)^2, {k, 0, n}], {n, 0, 15}]
Table[(n!)^2 SeriesCoefficient[1/(Exp[x] + Exp[y] - Exp[x + y])^n, {x, 0, n}, {y, 0, n}], {n, 0, 15}]
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a(n) = sum(k=0, n, binomial(n+k-1, k)*(k!*stirling(n, k, 2))^2); \\ Seiichi Manyama, Apr 06 2025
Showing 1-3 of 3 results.