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A161407 Number of partitions of n^2 into parts smaller than n.

%I A161407 #16 Sep 08 2021 07:09:21
%S A161407 1,0,1,5,30,185,1226,8442,60289,442089,3314203,25295011,195990980,
%T A161407 1538069121,12203218743,97746332667,789480879664,6423539487002,
%U A161407 52607252796831,433368610079872,3588859890833443,29862449600982149,249560820679038935,2093852201126089073
%N A161407 Number of partitions of n^2 into parts smaller than n.
%H A161407 Alois P. Heinz, <a href="/A161407/b161407.txt">Table of n, a(n) for n = 0..320</a>
%F A161407 a(n) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.0881548837986971165169272782933415... - _Vaclav Kotesovec_, Sep 08 2021
%e A161407 a(3) = #{2+2+2+2+1, 2+2+2+1+1+1, 2+2+5x1, 2+7x1, 9x1} = 5.
%p A161407 a := proc (n) local G, Gser: G := 1/(product(1-x^j, j = 1 .. n-1)): Gser := series(G, x = 0, n^2+5): coeff(Gser, x, n^2) end proc: 1, seq(a(n), n = 1 .. 23); # _Emeric Deutsch_, Jun 20 2009
%p A161407 # second Maple program:
%p A161407 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A161407      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
%p A161407     end:
%p A161407 a:= n-> b(n^2, n-1):
%p A161407 seq(a(n), n=0..30);  # _Alois P. Heinz_, Dec 21 2014
%t A161407 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[n^2, n-1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 30 2015, after _Alois P. Heinz_ *)
%Y A161407 Cf. A072213, A093115, A109655, A161408.
%K A161407 nonn
%O A161407 0,4
%A A161407 _Reinhard Zumkeller_, Jun 10 2009
%E A161407 More terms from _Emeric Deutsch_, Jun 20 2009
%E A161407 a(0)=1 from _Alois P. Heinz_, Dec 21 2014