cp's OEIS Frontend

A188216 Expansion of 1 + Sum_{n>=1} (x^(n^2) / Product_{k>=n} (1 - x^k)).

%I A188216 #20 Jan 26 2021 16:22:37
%S A188216 1,1,1,2,4,5,8,12,17,25,34,46,64,86,114,151,200,258,335,431,552,703,
%T A188216 891,1121,1411,1764,2196,2725,3374,4155,5111,6260,7650,9319,11329,
%U A188216 13726,16608,20031,24114,28962,34725,41529,49595,59095,70304,83476,98968,117109
%N A188216 Expansion of 1 + Sum_{n>=1} (x^(n^2) / Product_{k>=n} (1 - x^k)).
%C A188216 Number of partitions of n such that if k is the least part, then k occurs at least k times. - _Joerg Arndt_, Apr 17 2011
%H A188216 Alois P. Heinz, <a href="/A188216/b188216.txt">Table of n, a(n) for n = 0..10000</a>
%p A188216 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A188216      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
%p A188216     end:
%p A188216 a:= n-> `if`(n=0, 1, add(b(n-j^2, j), j=1..isqrt(n))):
%p A188216 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jan 03 2021
%t A188216 b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
%t A188216 a[n_] := If[n==0, 1, Sum[b[n - j^2, j], {j, 1, Sqrt[n]}]];
%t A188216 a /@ Range[0, 50] (* _Jean-François Alcover_, Jan 25 2021, after _Alois P. Heinz_ *)
%o A188216 (PARI) N=55; x='x+O('x^N);
%o A188216 t=1+sum(n=1,N,x^(n^2)/prod(k=n,N,1-x^k));
%o A188216 Vec(t)
%Y A188216 Cf. A096403 (expansion of sum(n>=1, x^(n^2) / prod(k>=n+1, 1-x^k)) ).
%Y A188216 Cf. A003114 (largest part k occurs at least k times).
%K A188216 nonn
%O A188216 0,4
%A A188216 _Joerg Arndt_, Mar 24 2011