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

0,3

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.

Links

Table of n, a(n) for n=0..40.

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 + ....

Example

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.

Sequence in context: A021993 A324743 A240011 * A338463 A186505 A228693

Adjacent sequences: A211225 A211226 A211227 * A211229 A211230 A211231

Keyword

nonn,easy

Author

Peter Bala, Apr 05 2012

Status

approved