A328301 Expansion of Product_{k>0} 1/(1 - x^(k^k)).

1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 9, 9, 9, 10, 11, 11, 11, 12, 13, 13, 13, 14, 15, 15, 15, 16, 17, 17, 17, 18, 19, 19, 19, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39

Offset

0,5

Comments

Also number of partitions of n into parts k^k for k > 0.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Formula

G.f.: 1 + Sum_{n>0} x^(n^n) / Product_{k=1..n} (1 - x^(k^k)).

Example

G.f.: 1 + x/(1-x) + x^4/((1-x)*(1-x^4)) + x^27/((1-x)*(1-x^4)*(1-x^27)) + ... .

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      b(n, i-1)+(p-> `if`(p>n, 0, b(n-p, i)))(i^i))
    end:
a:= n-> `if`(n<2, 1, b(n, floor((t-> t/LambertW(t))(log(n))))):
seq(a(n), n=0..100);  # Alois P. Heinz, Oct 12 2019

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1, b[n, i - 1] + With[{p = i^i}, If[p > n, 0, b[n - p, i]]]];
a[n_] := If[n < 2, 1, b[n, Floor[PowerExpand[Log[n]/ProductLog[Log[n]]]]]];
a /@ Range[0, 100] (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)

Prog

(PARI) my(N=99, x='x+O('x^N)); m=1; while(N>=m^m, m++); Vec(1/prod(k=1, m-1, 1-x^k^k))

Crossrefs

Cf. A000312, A001156, A003108, A046042, A064986, A328325.

Sequence in context: A257839 A075245 A367329 * A129253 A389975 A008652

Adjacent sequences: A328298 A328299 A328300 * A328302 A328303 A328304

Keyword

nonn

Author

Seiichi Manyama, Oct 12 2019

Status

approved