A306022
Stirling transform of partitions numbers (A000041).
Original entry on oeis.org
1, 1, 3, 10, 38, 163, 774, 4006, 22376, 133951, 854402, 5775948, 41190317, 308651432, 2422315371, 19856073597, 169596622997, 1506139073454, 13879704561038, 132488897335228, 1307829322689944, 13330635710335512, 140118664473276174, 1516899115597189064
Offset: 0
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a:= n-> add(combinat[numbpart](j)*Stirling2(n, j), j=0..n):
seq(a(n), n=0..30); # Alois P. Heinz, Jun 17 2018
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Table[Sum[StirlingS2[n, k]*PartitionsP[k], {k, 0, n}], {n, 0, 25}]
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a(n) = sum(k=0, n, stirling(n, k, 2)*numbpart(k)); \\ Michel Marcus, Jun 17 2018
A298905
Expansion of e.g.f. Product_{k>=1} (1 + log(1 + x)^k).
Original entry on oeis.org
1, 1, 1, 8, -8, 224, -712, 9120, -53496, 980088, -14394648, 264140832, -4113747024, 59028225840, -545558201424, -4191307074432, 450100910950272, -17302659472138752, 530508727766191104, -14790496500550616832, 408513443917280375808, -12274212131738107257600
Offset: 0
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b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
a:= n-> add(Stirling1(n, j)*b(j)*j!, j=0..n):
seq(a(n), n=0..23); # Alois P. Heinz, Jun 18 2018
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nmax = 21; CoefficientList[Series[Product[(1 + Log[1 + x]^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) Log[1 + x]^k/(k (1 - Log[1 + x]^k)), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] PartitionsQ[k] k!, {k, 0, n}], {n, 0, 21}]
A316146
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000009(k) * k^k.
Original entry on oeis.org
1, 5, 67, 865, 15906, 365514, 9545026, 276368635, 9188742238, 343857717788, 13998751394662, 618098575755637, 29469995998980356, 1510585321262760900, 83100039017148288635, 4873627957977247842223, 302388593396139280682588, 19804146883678522219587314
Offset: 1
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Table[Sum[StirlingS2[n, k] * PartitionsQ[k] * k^k, {k, 1, n}], {n, 1, 20}]
Showing 1-3 of 3 results.