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A123639 Consider the 2^n compositions of n and count only those ending in an even part. 4
0, 1, 2, 6, 18, 61, 224, 890, 3784, 17113, 81950, 414230, 2204110, 12314109, 72049548, 440379770, 2805266692, 18584809833, 127812870474, 910990458022, 6719535098378, 51223251471453, 403044829472760, 3269538955148698, 27314067026782976, 234749040898160153 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Compositions ending in an even part yield sequence 0 1 2 6 18 ... (this sequence). and A123638(n)+a(n) = A047970(n). Ending parity of compositions can be detected using mod(A065120,2)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..225

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 1+1+4+0 even compositions therefore a(4)=6.

MAPLE

g:= proc(b, t, l, m) option remember; if t=0 then b*(1-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); # Alois P. Heinz, Nov 06 2009

MATHEMATICA

g[b_, t_, l_, m_] := g[b, t, l, m] = If[ t == 0 , b*(1-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-Fran├žois Alcover, Nov 04 2013, translated from Alois P. Heinz's Maple program *)

CROSSREFS

Cf. A001045, A047970, A065120, A123638, A123640, A123641.

Sequence in context: A192483 A150051 A148462 * A228448 A177473 A177471

Adjacent sequences:  A123636 A123637 A123638 * A123640 A123641 A123642

KEYWORD

nonn

AUTHOR

Alford Arnold, Oct 04 2006

EXTENSIONS

More terms from Alois P. Heinz, Nov 06 2009

STATUS

approved

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Last modified February 23 23:00 EST 2018. Contains 299595 sequences. (Running on oeis4.)