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

0,3

Links

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

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

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 *)

Prog

(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), ", "))

Crossrefs

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

Sequence in context: A377506 A241839 A218183 * A091643 A323620 A306183

Adjacent sequences: A206224 A206225 A206226 * A206228 A206229 A206230

Keyword

nonn

Author

Paul D. Hanna, Feb 05 2012

Status

approved