A206227 Number of partitions of n^2+n into parts not greater than n.

1, 1, 4, 19, 108, 674, 4494, 31275, 225132, 1662894, 12541802, 96225037, 748935563, 5900502806, 46976736513, 377425326138, 3056671009814, 24930725879856, 204623068332997, 1688980598900228, 14012122025369431, 116784468316023069, 977437078888272796, 8212186058546599006

Offset: 0

Author

Paul D. Hanna, Feb 05 2012

Keywords

• nonn

Links

• Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..382 (first 150 terms from Alois P. Heinz)

Crossrefs

Cf. A173519, A206226, A206240, A107379, A258268.

Programs

Maple

T:= proc(n, k) option remember;
      `if`(n=0 or k=1, 1, T(n, k-1) + `if`(k>n, 0, T(n-k, k)))
    end:
seq(T(n^2+n, n), n=0..20); # Vaclav Kotesovec, May 25 2015 after Alois P. Heinz

Mathematica

Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,n*(n+1)}],{n,0,20}] (* Vaclav Kotesovec, May 25 2015 *)

PARI

{a(n)=polcoeff(prod(k=1,n,1/(1-x^k+x*O(x^(n^2+n)))),n^2+n)}
for(n=0,30,print1(a(n),", "))

Formula

a(n) = [x^(n^2+n)] Product_{k=1..n} 1/(1 - x^k).

a(n) ~ c * d^n / n^2, where d = 9.1533701924541224619485302924013545... = A258268, c = 0.3572966225745094270279188015952797... . - Vaclav Kotesovec, Sep 07 2014