A294982 Number of compositions (ordered partitions) of 1 into exactly 3n+1 powers of 1/(n+1).

1, 13, 217, 4245, 90376, 2019836, 46570140, 1097525253, 26293568950, 638048716305, 15643738390215, 386826618273420, 9633468179090952, 241366000080757480, 6078975012187601768, 153798067122829610085, 3906583987216447704594, 99579591801208823965115

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..700

Formula

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

Example

a(0) = 1: [1].
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].

Maple

a:= proc(n) option remember; `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))
    end:
seq(a(n), n=0..20);

Mathematica

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)];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, May 21 2018, translated from Maple *)
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 *)

Crossrefs

Row n=3 of A294746.

Sequence in context: A140517 A096141 A218475 * A320627 A059525 A086147

Adjacent sequences: A294979 A294980 A294981 * A294983 A294984 A294985

Keyword

nonn

Author

Alois P. Heinz, Nov 12 2017

Status

approved