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
Keywords
Examples
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)) + ... .
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..10000
Programs
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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 *) -
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))
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)).
Comments