A306023
Stirling transform of partitions into distinct parts (A000009).
Original entry on oeis.org
1, 1, 2, 6, 22, 89, 391, 1875, 9822, 55817, 340535, 2208681, 15118109, 108677575, 817914056, 6431115486, 52741729600, 450432487463, 3999401133601, 36853795902353, 351799243932131, 3472526583025397, 35382850151528847, 371592232539942447, 4016792440158613798
Offset: 0
-
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(b(j)*Stirling2(n, j), j=0..n):
seq(a(n), n=0..30); # Alois P. Heinz, Jun 17 2018
-
Table[Sum[StirlingS2[n, k]*PartitionsQ[k], {k, 0, n}], {n, 0, 25}]
A316145
a(n) = Sum_{k=0..n} Stirling2(n,k) * A000041(k) * k^k.
Original entry on oeis.org
1, 9, 106, 1823, 36821, 932080, 26666067, 876727561, 32137538059, 1305168046976, 57774609056649, 2783202675369037, 144453227105110782, 8035192765567735275, 476686201707606976317, 30053582893540865299197, 2005019178999976881804130, 141111387620531900621281975
Offset: 1
-
Table[Sum[StirlingS2[n, k] * PartitionsP[k] * k^k, {k, 1, n}], {n, 1, 20}]
A327601
Expansion of e.g.f. exp(x) * Product_{k>=1} 1/(1 - (1 - exp(x))^k).
Original entry on oeis.org
1, 0, 2, 0, 26, 120, 1922, 21840, 307946, 4251240, 63165842, 1010729280, 18501318266, 391496665560, 9265945721762, 232411950454320, 5972325812958986, 156131611764907080, 4208451299935189682, 119669466221148348960, 3658459009408581118106
Offset: 0
-
nmax = 20; CoefficientList[Series[Exp[x] Product[1/(1 - (1 - Exp[x])^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^k StirlingS2[n + 1, k + 1] k! PartitionsP[k], {k, 0, n}], {n, 0, 20}]
-
a(n) = sum(k=0, n, (-1)^k*stirling(n+1, k+1, 2)*k!*numbpart(k)); \\ Michel Marcus, Sep 19 2019
Showing 1-3 of 3 results.
Comments