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A304134 Number of partitions of 5n into exactly n parts.

%I A304134 #42 Nov 27 2020 02:08:01
%S A304134 1,1,5,19,64,192,532,1367,3319,7657,16928,36043,74287,148702,290071,
%T A304134 552767,1031391,1887776,3395084,6007963,10474462,18010859,30574655,
%U A304134 51284587,85064661,139620591,226914505,365371100,583164222,923075291,1449643115,2259616844
%N A304134 Number of partitions of 5n into exactly n parts.
%C A304134 Also, the number of partitions of 4n in which every part is <=n.
%H A304134 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 A304134 n | Partitions of 5n into exactly n parts
%e A304134 --+------------------------------------------------
%e A304134 1 | 5;
%e A304134 2 | 9+1, 8+2, 7+3, 6+4, 5+5;
%e A304134 3 | 13+1+1, 12+2+1, 11+3+1, 11+2+2, 10+4+1, 10+3+2,
%e A304134   |  9+5+1,  9+4+2,  9+3+3,  8+6+1,  8+5+2,  8+4+3,
%e A304134   |  7+7+1,  7+6+2,  7+5+3,  7+4+4,  6+6+3,  6+5+4,
%e A304134   |  5+5+5;
%e A304134 ====================================================================
%e A304134 n | Partitions of 4n in which every part is <=n.
%e A304134 --+-----------------------------------------------------------------
%e A304134 1 | 1+1+1+1;
%e A304134 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 A304134 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 A304134   | 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 A304134   | 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 A304134   | 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 A304134   | 2+2+1+1+1+1+1+1+1+1, 2+1+1+1+1+1+1+1+1+1+1,
%e A304134   | 1+1+1+1+1+1+1+1+1+1+1+1;
%p A304134 b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p A304134       b(n, i-1) +b(n-i, min(i, n-i)))
%p A304134     end:
%p A304134 a:= n-> b(4*n, n):
%p A304134 seq(a(n), n=0..35);  # _Alois P. Heinz_, May 07 2018
%t A304134 b[n_, i_] := b[n, i] = If[n==0 || i==1, 1, b[n, i-1] + b[n-i, Min[i, n-i]]];
%t A304134 a[n_] := b[4n, n];
%t A304134 a /@ Range[0, 35] (* _Jean-François Alcover_, Nov 27 2020, after _Alois P. Heinz_ *)
%o A304134 (PARI) {a(n) = polcoeff(prod(k=1, n, 1/(1-x^k+x*O(x^(4*n)))), 4*n)}
%Y A304134 Cf. A209816, A209818.
%K A304134 nonn
%O A304134 0,3
%A A304134 _Seiichi Manyama_, May 07 2018
%E A304134 More terms from _Alois P. Heinz_, May 07 2018