A160565 Diagonal sums of number triangle [k<=n]*C(n,2n-2k)2^(n-k)A000108(n-k).
1, 0, 1, 2, 1, 6, 9, 12, 41, 60, 121, 310, 505, 1162, 2577, 4760, 11089, 23256, 47089, 107274, 223345, 476366, 1061017, 2237796, 4888313, 10745748, 23048169, 50792638, 111180265, 241786898, 534219297
Offset: 0
Crossrefs
Cf.: A025250.
Formula
G.f.: (1-x^2-sqrt(1-2x^2-8x^3+x^4))/(4x^3);
G.f.: 1/(1-x^2-2*x^3/(1-x^2-2*x^3/(1-x^2-2*x^3/(1-x^2-2*x^3/(1-... (continued fraction).
a(n)=sum{k=0..floor(n/2), C(n-k,2n-4k)*2^(n-2k)*A000108(n-2k)};
a(n)=sum{k=0..n, C(n-k/2,2(n-k))*2^(n-k)*A000108(n-k)*(1+(-1)^k)/2};
a(n)=sum{k=0..n, C((n+k)/2,2k)*2^k*A000108(k)(1+(-1)^(n-k))/2}.
G.f.: (1/(1-x^2))c(2x^3/(1-x^2)^2) where c(x) is the g.f. of A000108. [From Paul Barry, May 20 2009]
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