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A304212 Number of partitions of n^3 into exactly n^2 parts. 2
1, 1, 5, 318, 112540, 139620591, 491579082022, 4303961368154069, 85434752794871493882, 3588523098005804563697043, 302194941264401427042462944147, 48844693123353655726678707534158535, 14615188708581196626576773497618986350642 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Chai Wah Wu, Table of n, a(n) for n = 0..50 (terms n = 0..30 from Seiichi Manyama)

FORMULA

a(n) = [x^(n^3-n^2)] Product_{k=1..n^2} 1/(1-x^k).

EXAMPLE

n | Partitions of n^3 into exactly n^2 parts

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

1 | 1.

2 | 5+1+1+1 = 4+2+1+1 = 3+3+1+1 = 3+2+2+1 = 2+2+2+2.

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^2, n^2):

seq(a(n), n=0..15);  # Alois P. Heinz, May 08 2018

MATHEMATICA

$RecursionLimit = 2000;

b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1]+b[n-i, Min[i, n-i]]];

a[n_] := b[n^3 - n^2, n^2]; a /@ Range[0, 15] (* Jean-Fran├žois Alcover, Nov 15 2020, after Alois P. Heinz *)

PROG

(PARI) {a(n) = polcoeff(prod(k=1, n^2, 1/(1-x^k+x*O(x^(n^3-n^2)))), n^3-n^2)}

(Python)

import sys

from functools import lru_cache

sys.setrecursionlimit(10**6)

@lru_cache(maxsize=None)

def b(n, i): return 1 if n == 0 or i == 1 else b(n, i-1)+b(n-i, min(i, n-i))

def A304212(n): return b(n**3-n**2, n**2) # Chai Wah Wu, Sep 09 2021, after Alois P. Heinz

CROSSREFS

Cf. A128854, A304176.

Sequence in context: A160071 A094161 A130308 * A212041 A251696 A274306

Adjacent sequences:  A304209 A304210 A304211 * A304213 A304214 A304215

KEYWORD

nonn

AUTHOR

Seiichi Manyama, May 08 2018

STATUS

approved

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Last modified October 16 20:36 EDT 2021. Contains 348047 sequences. (Running on oeis4.)