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Number of partitions of 5n into exactly n parts.
1

%I #42 Nov 27 2020 02:08:01

%S 1,1,5,19,64,192,532,1367,3319,7657,16928,36043,74287,148702,290071,

%T 552767,1031391,1887776,3395084,6007963,10474462,18010859,30574655,

%U 51284587,85064661,139620591,226914505,365371100,583164222,923075291,1449643115,2259616844

%N Number of partitions of 5n into exactly n parts.

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

%H Alois P. Heinz, <a href="/A304134/b304134.txt">Table of n, a(n) for n = 0..3000</a> (first 501 terms from Seiichi Manyama)

%e n | Partitions of 5n into exactly n parts

%e --+------------------------------------------------

%e 1 | 5;

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

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

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

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

%e | 5+5+5;

%e ====================================================================

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

%e --+-----------------------------------------------------------------

%e 1 | 1+1+1+1;

%e 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;

%e 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,

%e | 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,

%e | 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,

%e | 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,

%e | 2+2+1+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1+1,

%e | 1+1+1+1+1+1+1+1+1+1+1+1;

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

%p b(n, i-1) +b(n-i, min(i, n-i)))

%p end:

%p a:= n-> b(4*n, n):

%p seq(a(n), n=0..35); # _Alois P. Heinz_, May 07 2018

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

%t a[n_] := b[4n, n];

%t a /@ Range[0, 35] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)

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

%Y Cf. A209816, A209818.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 07 2018

%E More terms from _Alois P. Heinz_, May 07 2018