A211228 Shallow diagonal sums of A211226.

1, 1, 2, 2, 3, 4, 5, 8, 8, 15, 13, 28, 21, 51, 34, 92, 55, 164, 89, 290, 144, 509, 233, 888, 377, 1541, 610, 2662, 987, 4580, 1597, 7852, 2584, 13419, 4181, 22868, 6765, 38871, 10946, 65920, 17711

Offset: 0

Author

Peter Bala, Apr 05 2012

Keywords

• nonn
• easy

Comments

The even-indexed terms a(2*n) count the compositions of n+2 into odd parts while the odd-indexed terms a(2*n+3) count the total number of parts in the compositions of n+2 into odd parts.

Examples

			The compositions of 5 into odd parts are 1+1+1+1+1, 1+1+3, 1+3+1, 3+1+1 and 5. Hence a(6) = 5 and a(9) = 15.
		

Crossrefs

Cf. A000045, A029907, A211226.

Formula

Let f(n) := (floor(n/2))! and define c(n,k) = f(n)/(f(k)*f(n-k)) = A211226(n,k). Then a(n) = sum {k = 0..floor(n/2)} c(n-k,k).

a(2*n) = A000045(n+2); a(2*n-1) = A029907(n).

O.g.f.: (1+x-2*x^4-x^5-x^6)/(1-x^2-x^4)^2 = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + ....