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

0,3

Comments

Average of binomial and inverse binomial transforms of the Catalan numbers A000108.

Links

Table of n, a(n) for n=0..23.

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

Crossrefs

Cf. A007317, A086619.

Sequence in context: A319787 A148844 A145760 * A102206 A192856 A110886

Adjacent sequences: A102315 A102316 A102317 * A102319 A102320 A102321

Keyword

easy,nonn

Author

Paul Barry, Jan 04 2005

Status

approved