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

0,3

Comments

Exponential transform of A000009.

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

Formula

E.g.f.: exp(Sum_{k>=1} A000009(k)*x^k/k!).

Example

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! + ...

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}]

Crossrefs

Cf. A000009, A293839, A293840, A300511, A300515.

Sequence in context: A357798 A027221 A346747 * A150183 A150184 A150185

Adjacent sequences: A300511 A300512 A300513 * A300515 A300516 A300517

Keyword

nonn

Author

Ilya Gutkovskiy, Mar 07 2018

Status

approved