A206240 Number of partitions of n^2-n into parts not greater than n.

1, 1, 2, 7, 34, 192, 1206, 8033, 55974, 403016, 2977866, 22464381, 172388026, 1341929845, 10573800028, 84192383755, 676491536028, 5479185281572, 44692412971566, 366844007355202, 3028143252035976, 25123376972033392, 209401287806758273, 1752674793617241002

Offset: 0

Author

Paul D. Hanna, Feb 05 2012

Keywords

• nonn

Comments

Also the number of partitions of n^2 into exactly n parts. - Seiichi Manyama, May 07 2018

Examples

			From _Seiichi Manyama_, May 07 2018: (Start)
n | Partitions of n^2 into exactly n parts
--+-------------------------------------------------------
1 | 1.
2 | 3+1 = 2+2.
3 | 7+1+1 = 6+2+1 = 5+3+1 = 5+2+2 = 4+4+1 = 4+3+2 = 3+3+3. (End)
		

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, A206227, 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 = 9.153370192454122461948530292401354540073... = A258268, c = 0.07005383646855329845970382163053268... . - Vaclav Kotesovec, Sep 07 2014