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%I #17 Sep 20 2019 04:06:15
%S 1,13,217,4245,90376,2019836,46570140,1097525253,26293568950,
%T 638048716305,15643738390215,386826618273420,9633468179090952,
%U 241366000080757480,6078975012187601768,153798067122829610085,3906583987216447704594,99579591801208823965115
%N Number of compositions (ordered partitions) of 1 into exactly 3n+1 powers of 1/(n+1).
%H Alois P. Heinz, <a href="/A294982/b294982.txt">Table of n, a(n) for n = 0..700</a>
%F For n>0, a(n) = binomial(3*n+1, n+1)*binomial(2*n, n) + binomial(3*n+1, n-1). - _Vaclav Kotesovec_, Sep 20 2019
%e a(0) = 1: [1].
%e a(1) = 13: [1/4,1/4,1/4,1/4], [1/2,1/4,1/8,1/8], [1/2,1/8,1/4,1/8], [1/2,1/8,1/8,1/4], [1/4,1/2,1/8,1/8], [1/4,1/8,1/2,1/8], [1/4,1/8,1/8,1/2], [1/8,1/2,1/4,1/8], [1/8,1/2,1/8,1/4], [1/8,1/4,1/2,1/8], [1/8,1/4,1/8,1/2], [1/8,1/8,1/2,1/4], [1/8,1/8,1/4,1/2].
%p a:= proc(n) option remember; `if`(n<2, 12*n+1, (3*n-1)*(3*n+1)*
%p 3*((15*n^3-31*n^2-4*n+8)*n*a(n-1)-3*(3*n-4)*(3*n-2)*
%p (3*n^2-2*n-2)*a(n-2))/((n+1)*(4*n+2)*(3*n^2-8*n+3)*n^2))
%p end:
%p seq(a(n), n=0..20);
%t a[n_] := a[n] = If[n < 2, 12*n + 1, (3*n - 1)*(3*n + 1)*3*((15*n^3 - 31*n^2 - 4*n + 8)*n*a[n-1] - 3*(3*n - 4)*(3*n - 2)*(3*n^2 - 2*n - 2)*a[n-2])/((n + 1)*(4*n + 2)*(3*n^2 - 8*n + 3)*n^2)];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 21 2018, translated from Maple *)
%t Table[Binomial[3*n + 1, n + 1]*Binomial[2*n, n] + Binomial[3*n + 1, n - 1], {n, 0, 20}] (* _Vaclav Kotesovec_, Sep 20 2019 *)
%Y Row n=3 of A294746.
%K nonn
%O 0,2
%A _Alois P. Heinz_, Nov 12 2017
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