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A304176 Number of partitions of n^3 into exactly n parts. 3
1, 1, 4, 61, 1906, 91606, 6023602, 505853354, 51900711796, 6306147384659, 886745696653253, 141778041323736643, 25417656781153090889, 5052180112449982704619, 1103058286595668300801794, 262487324530101028337614478, 67628783852463631751658038290 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..96

FORMULA

a(n) = [x^(n^3-n)] Product_{k=1..n} 1/(1-x^k).

EXAMPLE

n | Partitions of n^3 into exactly n parts

--+------------------------------------------------------------

1 |   1.

2 |   7+1 = 6+2 = 5+3 = 4+4.

3 |   25+ 1+1 = 24+ 2+1 = 23+ 3+1 = 23+ 2+2 = 22+ 4+1 = 22+ 3+2

  | = 21+ 5+1 = 21+ 4+2 = 21+ 3+3 = 20+ 6+1 = 20+ 5+2 = 20+ 4+3

  | = 19+ 7+1 = 19+ 6+2 = 19+ 5+3 = 19+ 4+4 = 18+ 8+1 = 18+ 7+2

  | = 18+ 6+3 = 18+ 5+4 = 17+ 9+1 = 17+ 8+2 = 17+ 7+3 = 17+ 6+4

  | = 17+ 5+5 = 16+10+1 = 16+ 9+2 = 16+ 8+3 = 16+ 7+4 = 16+ 6+5

  | = 15+11+1 = 15+10+2 = 15+ 9+3 = 15+ 8+4 = 15+ 7+5 = 15+ 6+6

  | = 14+12+1 = 14+11+2 = 14+10+3 = 14+ 9+4 = 14+ 8+5 = 14+ 7+6

  | = 13+13+1 = 13+12+2 = 13+11+3 = 13+10+4 = 13+ 9+5 = 13+ 8+6

  | = 13+ 7+7 = 12+12+3 = 12+11+4 = 12+10+5 = 12+ 9+6 = 12+ 8+7

  | = 11+11+5 = 11+10+6 = 11+ 9+7 = 11+ 8+8 = 10+10+7 = 10+ 9+8

  | =  9+ 9+9.

MAPLE

b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,

      b(n, i-1)+b(n-i, min(i, n-i)))

    end:

a:= n-> b(n^3-n, n):

seq(a(n), n=0..20);  # Alois P. Heinz, May 07 2018

PROG

(PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(n^3-n)))), n^3-n)}

CROSSREFS

Cf. A206240, A238608, A304208, A304212.

Sequence in context: A252973 A262654 A218472 * A326225 A102193 A232156

Adjacent sequences:  A304173 A304174 A304175 * A304177 A304178 A304179

KEYWORD

nonn

AUTHOR

Seiichi Manyama, May 07 2018

STATUS

approved

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Last modified May 30 05:35 EDT 2020. Contains 334712 sequences. (Running on oeis4.)