A300512 Expansion of e.g.f. log(Sum_{k>=0} p(k)*x^k/k!), where p(k) = number of partitions of k (A000041).

0, 1, 1, -1, -1, 6, -1, -77, 203, 1344, -10692, -15862, 579611, -1518768, -32884753, 283168220, 1550435633, -38615194078, 44538307279, 4920513118440, -39485852954288, -546206846420721, 11322395643617278, 23746787652752639, -2713550731461618505, 17064642256532964421

Offset: 0

Author

Ilya Gutkovskiy, Mar 07 2018

Keywords

• sign

Comments

Logarithmic transform of A000041.

Examples

			E.g.f.: A(x) = x/1! + x^2/2! - x^3/3! - x^4/4! + 6*x^5/5! - x^6/6! - 77*x^7/7! + 203*x^8/8! + ...
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..483
• N. J. A. Sloane, Transforms
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000041, A038048, A300511.

Programs

Maple

a:= proc(n) option remember; (t-> `if`(n=0, 0, t(n)-add(j*a(j)*
      binomial(n, j)*t(n-j), j=1..n-1)/n))(combinat[numbpart])
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 07 2018

Mathematica

nmax = 25; CoefficientList[Series[Log[Sum[PartitionsP[k] x^k/k!, {k, 0, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = PartitionsP[n] - Sum[k Binomial[n, k] PartitionsP[n - k] a[k], {k, 1, n - 1}]/n; a[0] = 0; Table[a[n], {n, 0, 25}]

Formula

E.g.f.: log(Sum_{k>=0} A000041(k)*x^k/k!).