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

1, 0, 1, 5, 30, 185, 1226, 8442, 60289, 442089, 3314203, 25295011, 195990980, 1538069121, 12203218743, 97746332667, 789480879664, 6423539487002, 52607252796831, 433368610079872, 3588859890833443, 29862449600982149, 249560820679038935, 2093852201126089073

Offset: 0

Author

Reinhard Zumkeller, Jun 10 2009

Keywords

• nonn

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..320

Crossrefs

Cf. A072213, A093115, A109655, A161408.

Programs

Maple

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
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(n^2, n-1):
seq(a(n), n=0..30);  # Alois P. Heinz, Dec 21 2014

Mathematica

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 *)

Formula

a(n) ~ c * d^n / n^2, where d = A258268 = 9.153370192454122461948530292401354... and c = 0.0881548837986971165169272782933415... - Vaclav Kotesovec, Sep 08 2021

Extensions

More terms from Emeric Deutsch, Jun 20 2009

a(0)=1 from Alois P. Heinz, Dec 21 2014