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A304134 Number of partitions of 5n into exactly n parts. 1
1, 1, 5, 19, 64, 192, 532, 1367, 3319, 7657, 16928, 36043, 74287, 148702, 290071, 552767, 1031391, 1887776, 3395084, 6007963, 10474462, 18010859, 30574655, 51284587, 85064661, 139620591, 226914505, 365371100, 583164222, 923075291, 1449643115, 2259616844 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Also, the number of partitions of 4n in which every part is <=n.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..3000 (first 501 terms from Seiichi Manyama)

EXAMPLE

n | Partitions of 5n into exactly n parts

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

1 | 5;

2 | 9+1, 8+2, 7+3, 6+4, 5+5;

3 | 13+1+1, 12+2+1, 11+3+1, 11+2+2, 10+4+1, 10+3+2,

| 9+5+1, 9+4+2, 9+3+3, 8+6+1, 8+5+2, 8+4+3,

| 7+7+1, 7+6+2, 7+5+3, 7+4+4, 6+6+3, 6+5+4,

| 5+5+5;

====================================================================

n | Partitions of 4n in which every part is <=n.

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

1 | 1+1+1+1;

2 | 2+2+2+2, 2+2+2+1+1, 2+2+1+1+1+1, 2+1+1+1+1+1+1, 1+1+1+1+1+1+1+1;

3 | 3+3+3+3, 3+3+3+2+1, 3+3+3+1+1+1, 3+3+2+2+2, 3+3+2+2+1+1,

| 3+3+2+1+1+1+1, 3+3+1+1+1+1+1+1, 3+2+2+2+2+1, 3+2+2+2+1+1+1,

| 3+2+2+1+1+1+1+1, 3+2+1+1+1+1+1+1+1, 3+1+1+1+1+1+1+1+1+1,

| 2+2+2+2+2+2, 2+2+2+2+2+1+1, 2+2+2+2+1+1+1+1, 2+2+2+1+1+1+1+1+1,

| 2+2+1+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1+1,

| 1+1+1+1+1+1+1+1+1+1+1+1;

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(4*n, n):

seq(a(n), n=0..35); # Alois P. Heinz, May 07 2018

MATHEMATICA

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[4n, n];

a /@ Range[0, 35] (* Jean-François Alcover, Nov 27 2020, after Alois P. Heinz *)

PROG

(PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(4*n)))), 4*n)}

CROSSREFS

Cf. A209816, A209818.

Sequence in context: A053545 A049612 A211842 * A318946 A229239 A296330

Adjacent sequences: A304131 A304132 A304133 * A304135 A304136 A304137

KEYWORD

nonn

AUTHOR

Seiichi Manyama, May 07 2018

EXTENSIONS

More terms from Alois P. Heinz, May 07 2018

STATUS

approved

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Last modified November 27 22:56 EST 2022. Contains 358406 sequences. (Running on oeis4.)