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

0,4

Links

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

Formula

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

Example

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

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 *)

Crossrefs

Cf. A072213, A093115, A109655, A161408.

Sequence in context: A147837 A076036 A195257 * A006773 A059273 A352175

Adjacent sequences: A161404 A161405 A161406 * A161408 A161409 A161410

Keyword

nonn

Author

Reinhard Zumkeller, Jun 10 2009

Extensions

More terms from Emeric Deutsch, Jun 20 2009

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

Status

approved