A211228 - OEIS

%I #8 Sep 20 2013 11:34:45

%S 1,1,2,2,3,4,5,8,8,15,13,28,21,51,34,92,55,164,89,290,144,509,233,888,

%T 377,1541,610,2662,987,4580,1597,7852,2584,13419,4181,22868,6765,

%U 38871,10946,65920,17711

%N Shallow diagonal sums of A211226.

%C 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.

%F 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).

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

%F 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 + ....

%e 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.

%Y Cf. A000045, A029907, A211226.

%K nonn,easy

%O 0,3

%A _Peter Bala_, Apr 05 2012