A102318 A mean binomial transform of the Catalan numbers.
1, 1, 3, 8, 27, 97, 373, 1493, 6163, 26027, 111897, 488006, 2153429, 9596199, 43121211, 195165576, 888861555, 4070582971, 18732710281, 86584519280, 401776434017, 1870983991035, 8740907398527, 40956401225597
Offset: 0
Formula
G.f.: (2-sqrt((1-3x)/(1+x))-sqrt((1-5x)/(1-x)))/(4x);
a(n)=sum{k=0..floor(n/2), binomial(n, 2k)C(n-2k)};
a(n)=sum{k=0..n, binomial(n, k)C(k)(1+(-1)^(n-k))/2}.
Conjecture: -(n-1)*(n+1)*a(n) +2*(5*n^2-9*n+1)*a(n-1) +2*(-15*n^2+58*n-49)*a(n-2) +2*(10*n^2-76*n+123)*a(n-3) +(31*n-55)*(n-3)*a(n-4) -30*(n-3)*(n-4)*a(n-5)=0. - R. J. Mathar, Jun 08 2016
Conjecture: +(3*n-10)*(n-1)*(n+1)*a(n) +2*(-12*n^3+58*n^2-67*n+10)*a(n-1) +2*(21*n^3-136*n^2+289*n-196)*a(n-2) +2*(n-2)*(12*n^2-46*n+27)*a(n-3) -15*(n-2)*(n-3)*(3*n-7)*a(n-4)=0. - R. J. Mathar, Jun 08 2016
a(n) ~ 5^(n + 3/2) / (16 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 30 2017
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