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 A102318 A mean binomial transform of the Catalan numbers. 0
 1, 1, 3, 8, 27, 97, 373, 1493, 6163, 26027, 111897, 488006, 2153429, 9596199, 43121211, 195165576, 888861555, 4070582971, 18732710281, 86584519280, 401776434017, 1870983991035, 8740907398527, 40956401225597 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 22 23:47 EDT 2024. Contains 373629 sequences. (Running on oeis4.)