A300514 Expansion of e.g.f. exp(Sum_{k>=1} q(k)*x^k/k!), where q(k) = number of partitions of k into distinct parts (A000009).
1, 1, 2, 6, 20, 79, 358, 1791, 9854, 58958, 379716, 2617320, 19197327, 149099827, 1221390172, 10515829901, 94865603724, 894302028718, 8788782784778, 89848652800152, 953666248076772, 10491219933196228, 119429574273909421, 1404835599743325765, 17052591331677804136
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x/1! + 2*x^2/2! + 6*x^3/3! + 20*x^4/4! + 79*x^5/5! + 358*x^6/6! + 1791*x^7/7! + ...
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..550
- N. J. A. Sloane, Transforms
- Index entries for related partition-counting sequences
Programs
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Maple
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:= proc(n) option remember; `if`(n=0, 1, add( a(n-j)*binomial(n-1, j-1)*b(j), j=1..n)) end: seq(a(n), n=0..30); # Alois P. Heinz, Mar 07 2018 -
Mathematica
nmax = 24; CoefficientList[Series[Exp[Sum[PartitionsQ[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]! a[n_] := a[n] = Sum[PartitionsQ[k] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 24}]
Formula
E.g.f.: exp(Sum_{k>=1} A000009(k)*x^k/k!).
Comments