A328325 Expansion of Product_{k>=0} 1/(1 - x^(k^k)).

1, 2, 3, 4, 6, 8, 10, 12, 15, 18, 21, 24, 28, 32, 36, 40, 45, 50, 55, 60, 66, 72, 78, 84, 91, 98, 105, 113, 122, 131, 140, 150, 161, 172, 183, 195, 208, 221, 234, 248, 263, 278, 293, 309, 326, 343, 360, 378, 397, 416, 435, 455, 476, 497, 519, 542, 566, 590, 615, 641, 668, 695

Offset

0,2

Comments

Partial sums of A328301.

Links

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

Formula

a(n) = Sum_{k=0..n} A328301(k).

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

Example

G.f.: 1/(1-x) + x/(1-x)^2 + x^4/((1-x)^2*(1-x^4)) + x^27/((1-x)^2*(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:= proc(n) option remember; `if`(n<2, n+1, a(n-1)+
      b(n, floor((t-> t/LambertW(t))(log(n)))))
    end:
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_] := a[n] = If[n < 2, n+1, a[n-1] + b[n, Floor[PowerExpand[Log[n]/ ProductLog[Log[n]]]]]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Sep 14 2022, after Alois P. Heinz *)

Prog

(PARI) N=99; x='x+O('x^N); m=1; while(N>=m^m, m++); Vec(1/prod(k=0, m-1, 1-x^k^k))

Crossrefs

Cf. A000312, A064985, A086858, A328301.

Sequence in context: A019293 A130519 A001972 * A005705 A139542 A347656

Adjacent sequences: A328322 A328323 A328324 * A328326 A328327 A328328

Keyword

nonn

Author

Seiichi Manyama, Oct 12 2019

Status

approved