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

Author

Paul D. Hanna, Jul 23 2004

Keywords

• nonn

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

Examples

			[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
		

Links

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

Crossrefs

Cf. A097084.

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

Programs

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 *)

Formula

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