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A052337 Number of partitions into at most a(1) copies of 1, a(2) copies of 2, ...

%I A052337 #14 Nov 23 2020 07:15:01
%S A052337 1,1,1,2,2,3,5,6,8,10,13,17,21,27,34,42,53,65,80,98,119,146,177,213,
%T A052337 258,309,370,443,528,628,746,883,1044,1231,1449,1703,1997,2338,2734,
%U A052337 3190,3718,4325,5025,5830,6754,7816,9032,10422,12016,13832,15907,18274
%N A052337 Number of partitions into at most a(1) copies of 1, a(2) copies of 2, ...
%H A052337 Alois P. Heinz, <a href="/A052337/b052337.txt">Table of n, a(n) for n = 0..10000</a>
%F A052337 E.g.f. satisfies A(x) = Product_{i>=1} (1-x^(a(i)*(i+1)))/(1-x^i).
%p A052337 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A052337       add(b(n-i*j, i-1), j=0..min(n/i, a(i)))))
%p A052337     end:
%p A052337 a:= n-> `if`(n=0, 0, 1)+b(n, n-1):
%p A052337 seq(a(n), n=0..70);  # _Alois P. Heinz_, Jun 12 2018
%t A052337 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, i - 1], {j, 0, Min[n/i, a[i]]}]]];
%t A052337 a[n_] := If[n == 0, 0, 1] + b[n, n - 1];
%t A052337 a /@ Range[0, 70] (* _Jean-François Alcover_, Nov 23 2020, after _Alois P. Heinz_ *)
%Y A052337 Cf. A289501.
%K A052337 nonn,eigen
%O A052337 0,4
%A A052337 _Christian G. Bower_, Dec 19 1999