A097085 - OEIS

%I #21 Jun 04 2018 03:12:17

%S 1,2,4,10,26,70,204,618,1908,6010,19316,63034,208210,695594,2346748,

%T 7983450,27364842,94439262,327922692,1145029314,4018618374,

%U 14169874350,50179643628,178410716622,636679332588,2279906714610,8190512723940

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

%C 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

%H Alois P. Heinz, <a href="/A097085/b097085.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{j=0..n} A263897(j). - _Gregory L. Simay_, Oct 30 2015

%e [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

%p b:= proc(n, i, p) option remember; `if`(n=0, p!^2,

%p       `if`(i<1, 0, add(b(n-i*j, i-1, p+j)/j!^2, j=0..n/i)))

%p     end:

%p a:= proc(n) option remember; b(n$2, 0)+`if`(n>0, a(n-1), 0) end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Oct 30 2015

%t 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}]]];

%t a[n_] := a[n] = b[n, n, 0] + If[n > 0, a[n - 1], 0];

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jun 04 2018, after _Alois P. Heinz_ *)

%Y Cf. A097084.

%Y Cf. A263897. - _Gregory L. Simay_, Oct 30 2015

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 23 2004