A097085 Row sums of triangle A097084, in which the n-th diagonal equals the n-th row transformed by triangle A008459 (squared binomial coefficients).

1, 2, 4, 10, 26, 70, 204, 618, 1908, 6010, 19316, 63034, 208210, 695594, 2346748, 7983450, 27364842, 94439262, 327922692, 1145029314, 4018618374, 14169874350, 50179643628, 178410716622, 636679332588, 2279906714610, 8190512723940

Offset

0,2

Comments

a(n) is also the number of anagram compositions of 2n or of 2n+1. A composition of n is an ordered sequence of positive integers whose sum is n. An anagram composition of n can be divided into two consecutive subsequences with exactly the same parts, with a central part between the subsequences permitted. - Gregory L. Simay, Oct 30 2015

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = Sum_{j=0..n} A263897(j). - Gregory L. Simay, Oct 30 2015

Example

[1,2,3,4][3,2,1,4] is an anagram composition of 20 enumerated by a(10), [3,2,1] 5 [2,1,3] is an anagram composition of 17 enumerated by a(8), [3467] 8 [7643] is an anagram composition of 48 enumerated by a(24). - Gregory L. Simay, Oct 30 2015

Maple

b:= proc(n, i, p) option remember; `if`(n=0, p!^2,
      `if`(i<1, 0, add(b(n-i*j, i-1, p+j)/j!^2, j=0..n/i)))
    end:
a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 30 2015

Mathematica

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!^2, If[i < 1, 0, Sum[b[n - i*j, i - 1, p + j]/j!^2, {j, 0, n/i}]]];
a[n_] := a[n] = b[n, n, 0] + If[n > 0, a[n - 1], 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 04 2018, after Alois P. Heinz *)

Crossrefs

Cf. A097084.

Cf. A263897. - Gregory L. Simay, Oct 30 2015

Sequence in context: A025565 A085455 A055226 * A071962 A257520 A035079

Adjacent sequences: A097082 A097083 A097084 * A097086 A097087 A097088

Keyword

nonn

Author

Paul D. Hanna, Jul 23 2004

Status

approved