%I #16 Sep 08 2021 07:09:21

%S 1,0,1,5,30,185,1226,8442,60289,442089,3314203,25295011,195990980,

%T 1538069121,12203218743,97746332667,789480879664,6423539487002,

%U 52607252796831,433368610079872,3588859890833443,29862449600982149,249560820679038935,2093852201126089073

%N Number of partitions of n^2 into parts smaller than n.

%H Alois P. Heinz, <a href="/A161407/b161407.txt">Table of n, a(n) for n = 0..320</a>

%F a(n) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.0881548837986971165169272782933415... - _Vaclav Kotesovec_, Sep 08 2021

%e a(3) = #{2+2+2+2+1, 2+2+2+1+1+1, 2+2+5x1, 2+7x1, 9x1} = 5.

%p 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 # second Maple program:

%p b:= proc(n, i) option remember; `if`(n=0, 1,

%p `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))

%p end:

%p a:= n-> b(n^2, n-1):

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Dec 21 2014

%t 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 Cf. A072213, A093115, A109655, A161408.

%K nonn

%O 0,4

%A _Reinhard Zumkeller_, Jun 10 2009

%E More terms from _Emeric Deutsch_, Jun 20 2009

%E a(0)=1 from _Alois P. Heinz_, Dec 21 2014