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

0,3

Comments

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

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

Example

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)

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, A206227, A107379, A258268.

Sequence in context: A058915 A273030 A020054 * A289720 A190631 A326560

Adjacent sequences: A206237 A206238 A206239 * A206241 A206242 A206243

Keyword

nonn

Author

Paul D. Hanna, Feb 05 2012

Status

approved