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A292507 Number of partitions of n with up to n distinct kinds of 1.

%I A292507 #21 Apr 25 2021 02:33:01
%S A292507 1,1,2,5,13,33,82,201,488,1176,2817,6714,15931,37647,88628,207914,
%T A292507 486158,1133304,2634339,6106953,14121157,32573842,74968044,172164086,
%U A292507 394561089,902471184,2060338222,4695324425,10681885697,24261437446,55017434305,124573678280
%N A292507 Number of partitions of n with up to n distinct kinds of 1.
%H A292507 Vaclav Kotesovec, <a href="/A292507/b292507.txt">Table of n, a(n) for n = 0..1382</a> (terms 0..1000 from Alois P. Heinz)
%F A292507 Conjecture: log(a(n)) ~ log(2)*n + Pi*sqrt(n/3) - 3*log(n)/2. - _Vaclav Kotesovec_, May 11 2019
%F A292507 a(n) = [x^n] (1 + x)^n * Product_{k>=2} 1 / (1 - x^k). - _Ilya Gutkovskiy_, Apr 24 2021
%e A292507 a(3) = 5: 3, 21a, 21b, 21c, 1a1b1c.
%e A292507 a(4) = 13: 4, 31a, 31b, 31c, 31d, 22, 21a1b, 21a1c, 21a1d, 21b1c, 21b1d, 21c1d, 1a1b1c1d.
%p A292507 b:= proc(n, i, k) option remember; `if`(n=0 or i=1,
%p A292507       binomial(k, n), `if`(i>n, 0, b(n-i, i, k))+b(n, i-1, k))
%p A292507     end:
%p A292507 a:= n-> b(n$3):
%p A292507 seq(a(n), n=0..35);
%t A292507 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, Binomial[k, n], If[i > n, 0, b[n - i, i, k]] + b[n, i - 1, k]];
%t A292507 a[n_] := b[n, n, n];
%t A292507 Table[a[n], {n, 0, 35}] (* _Jean-François Alcover_, May 20 2018, translated from Maple *)
%Y A292507 Main diagonal of A292622.
%Y A292507 Cf. A292462, A292463, A292503.
%K A292507 nonn
%O A292507 0,3
%A A292507 _Alois P. Heinz_, Sep 17 2017