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A206240 Number of partitions of n^2-n into parts not greater than n. 5
1, 1, 2, 7, 34, 192, 1206, 8033, 55974, 403016, 2977866, 22464381, 172388026, 1341929845, 10573800028, 84192383755, 676491536028, 5479185281572, 44692412971566, 366844007355202, 3028143252035976, 25123376972033392, 209401287806758273, 1752674793617241002 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 21 05:35 EST 2021. Contains 340333 sequences. (Running on oeis4.)