A211227 Row sums of A211226.

1, 2, 3, 4, 8, 8, 20, 16, 48, 32, 112, 64, 256, 128, 576, 256, 1280, 512, 2816, 1024, 6144, 2048, 13312, 4096, 28672, 8192, 61440, 16384, 131072, 32768, 278528, 65536, 589824, 131072, 1245184, 262144, 2621440, 524288, 5505024

Offset

0,2

Comments

The odd-indexed terms of the sequence a(2*n-1) count the compositions of n+1, while the even-indexed terms a(2*n) count the total number of parts in the composition of n+1. Compare with A211228.

Links

Table of n, a(n) for n=0..38.

Formula

a(n) = sum {k = 0..n } f(n)/(f(k)*f(n-k)), where f(n) := (floor(n/2))!.

a(2*n-1) = 2^n = A000079(n); a(2*n) = (n+2)*2^(n-1) = A001792(n).

O.g.f.: (1+2*x-x^2-4*x^3)/(1-2*x^2)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 8*x^4 + ....

E.g.f.: cosh(sqrt(2)*x) + (4+x)/(2*sqrt(2))*sinh(sqrt(2)*x) = 1 + 2*x + 3*x^2/2! + 4*x^3/3! + 8*x^4/4! + .....

Example

The four compositions of 3 are 1+1+1, 1+2, 2+1 and 3 having 8 parts in total. Hence a(3) = 4 and a(4) = 8.

Crossrefs

Cf. A000079, A001792, A211226.

Sequence in context: A242065 A056424 A050324 * A019949 A172002 A274927

Adjacent sequences: A211224 A211225 A211226 * A211228 A211229 A211230

Keyword

nonn,easy

Author

Peter Bala, Apr 05 2012

Status

approved