A192554
a(n) = Sum_{k=0..n} abs(Stirling1(n,k))*(-1)^(n-k)*k!^2.
Original entry on oeis.org
1, 1, 3, 26, 398, 9724, 344236, 16663968, 1056631824, 84962783664, 8446120969104, 1016998946575776, 145848462866589600, 24562489788256472064, 4799789988678066147840, 1077128972416478325901824, 275111625956753684599202304
Offset: 0
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Table[Sum[Abs[StirlingS1[n,k]](-1)^(n-k)k!^2,{k,0,n}],{n,0,100}]
Table[Sum[StirlingS1[n,k] * k!^2, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 05 2021 *)
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makelist(sum(abs(stirling1(n,k))*(-1)^(n-k)*k!^2,k,0,n),n,0,24);
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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
A351281
a(n) = Sum_{k=0..n} k! * k^k * Stirling2(n,k).
Original entry on oeis.org
1, 1, 9, 187, 7173, 440611, 39631509, 4910795107, 802015652853, 166948755155971, 43146953460348309, 13555255072473665827, 5087595330217093070133, 2248298922174973220446531, 1155512971750307157457879509, 683392198848998191062416885347
Offset: 0
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a[0] = 1; a[n_] := Sum[k! * k^k * StirlingS2[n, k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Feb 06 2022 *)
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a(n) = sum(k=0, n, k!*k^k*stirling(n, k, 2));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k*(exp(x)-1))^k)))
A351333
a(n) = Sum_{k=0..n} k! * (-k)^k * Stirling1(n,k).
Original entry on oeis.org
1, -1, 9, -188, 7210, -442534, 39778322, -4926514200, 804271290024, -167367096770256, 43244394345493968, -13583108127289832592, 5097183064576208028096, -2252211248747050526401296, 1157380447302779717382178416, -684423139836843936246492092928
Offset: 0
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a(n) = sum(k=0, n, k!*(-k)^k*stirling(n, k, 1));
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k*log(1+x))^k)))
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
Showing 1-4 of 4 results.