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

Author

Peter Bala, Apr 05 2012

Keywords

• nonn
• easy

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.

Examples

			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.

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! + .....