A123638 Consider the 2^n compositions of n and count only those ending in an odd part with row sum A001045.

1, 1, 3, 8, 25, 83, 299, 1158, 4813, 21373, 100955, 504916, 2662761, 14754311, 85643459, 519493938, 3285790317, 21628225041, 147887079907, 1048634836288, 7698589399833, 58432476430139, 457901993065915, 3700291495531166

Offset

1,3

Comments

Compositions ending in an even part yield sequence 0 1 2 6 18 ... A123639. and a(n)+A123639(n) = A047970(n). Ending parity of compositions can be detected using mod(A065120,2)

Links

Table of n, a(n) for n=1..24.

Example

4
31 32 33
211 221 222
1111
Consider the above multisets: permute and note the parity of the ending part of each of the 14 compositions.
4
31 13 32 23 33
211 121 112 221 212 122 222
1111
4 is even
31 13 23 and 33 are odd
32 is even
etc
there are 0 + 4 + 3 + 1 = 8 odd compositions therefore a(4)=8.

Maple

g:= proc(b, t, l, m) option remember; if t=0 then b*l else add (g(b, t-1, irem(k, 2), m), k=1..m-1) +g(1, t-1, irem(m, 2), m) fi end: a:= n-> add (g(0, k, 0, n+1-k), k=1..n): seq (a(n), n=1..30);

Mathematica

g[b_, t_, l_, m_] := g[b, t, l, m] = If[t == 0 , b*l , Sum[g[b, t-1, Mod[k, 2], m], {k, 1, m-1}] + g[1, t-1, Mod[m, 2], m]]; a[n_] := Sum[g[0, k, 0, n+1-k], {k, 1, n}]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 04 2013, translated from Alois P. Heinz's Maple program *)

Crossrefs

Cf. A001045 A047970 A065120 A123639 A123640 A123641.

Sequence in context: A148795 A148796 A148797 * A207651 A361959 A038665

Adjacent sequences: A123635 A123636 A123637 * A123639 A123640 A123641

Keyword

nonn

Author

Alford Arnold, Oct 04 2006

Extensions

Offset corrected, Maple program and more terms added by Alois P. Heinz, Nov 06 2009

Status

approved