A320502
a(n) = Sum_{k=0..n} (k!)^2 * abs(Stirling1(n,k)).
Original entry on oeis.org
1, 1, 5, 50, 842, 21644, 792676, 39297600, 2536525008, 206794669104, 20785423425264, 2525457805492896, 364910211591903072, 61847041340997089280, 12151693924459271926272, 2739901558132307387349504, 702704348810821821056454144, 203409730893592265642619623424
Offset: 0
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[(&+[Abs(StirlingFirst(n,k))*(Factorial(k))^2: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Oct 14 2018
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Table[Sum[Abs[StirlingS1[n, k]]*k!^2, {k, 0, n}], {n, 0, 20}]
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a(n) = sum(k=0, n, k!^2*abs(stirling(n, k, 1))); \\ Michel Marcus, Oct 14 2018
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, k!*(-log(1-x))^k))) \\ Seiichi Manyama, Apr 22 2022
A385752
a(n) = Sum_{k=0..n} Stirling1(n,k) * (n!/k!)^2.
Original entry on oeis.org
1, 1, -3, 46, -1967, 179351, -29861639, 8200834972, -3456505906559, 2118756407303197, -1811589861406160699, 2089746219541021377546, -3164800617505630505525903, 6151223064132377579849537011, -15052264342298428131766095419839, 45616620088948927404807879986431576, -168785206495071742797011703980958673919
Offset: 0
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Table[Sum[StirlingS1[n, k] (n!/k!)^2, {k, 0, n}], {n, 0, 16}]
nmax = 16; CoefficientList[Series[Sum[Log[1 + x]^k/k!^3, {k, 0, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^3
A382794
a(n) = Sum_{k=0..n} Stirling1(n,k) * Stirling2(n,k) * (k!)^2.
Original entry on oeis.org
1, 1, 3, 2, -418, -14676, -234344, 18565056, 2659703616, 169046742960, -6539356064736, -4061128974843744, -672969012637199040, -19289566159655581440, 27323548725052131528960, 10157639436460221570630144, 1433264952547826545065237504, -520046813680980959472490690560
Offset: 0
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Table[Sum[StirlingS1[n, k] StirlingS2[n, k] (k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 - (Exp[x] - 1) Log[1 + y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]
A382805
a(n) = Sum_{k=0..n} (-1)^(n-k) * (Stirling1(n,k) * k!)^2.
Original entry on oeis.org
1, 1, 3, 4, -272, -8524, -96596, 9634752, 983055168, 36429411456, -4303305703296, -1051644384152064, -89651253435644160, 10632887072757561600, 5599203549778990667520, 914684633796830925275136, -89559567563652079025946624, -104514775371103880549281775616
Offset: 0
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Table[Sum[(-1)^(n - k) (StirlingS1[n, k] k!)^2, {k, 0, n}], {n, 0, 17}]
Table[(n!)^2 SeriesCoefficient[1/(1 + Log[1 + x] Log[1 - y]), {x, 0, n}, {y, 0, n}], {n, 0, 17}]
Showing 1-4 of 4 results.