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 A270489 Sum_{k=0..n} ((binomial(3*k,k)*binomial(2*n-k,n))/(2*k+1)). 1
 1, 3, 12, 54, 265, 1401, 7903, 47088, 293319, 1892440, 12548041, 84988566, 585314652, 4085026386, 28820064810, 205156454376, 1471492171068, 10622954509803, 77122189800121, 562684397212060, 4123449352097229, 30336562360256955 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 FORMULA G.f.: P(C(x))/(1-x/(1-C(x)))^2/x, where C(x)=(1-sqrt(1-4*x))/2, P(x)/x is g.f. of A001764. Recurrence: 46*(n-2)*(n-1)*n*(2*n + 1)*(133*n - 239)*a(n) = 5*(n-2)*(n-1)*(38969*n^3 - 108996*n^2 + 78733*n - 18774)*a(n-1) - 4*(n-2)*(242858*n^4 - 1286417*n^3 + 2496793*n^2 - 2103937*n + 643755)*a(n-2) + 36*(3*n - 7)*(3*n - 5)*(6*n - 13)*(6*n - 11)*(133*n - 106)*a(n-3). - Vaclav Kotesovec, Mar 18 2016 a(n) ~ 3^(6*n + 7/2) / (19^(3/2) * sqrt(Pi) * 2^(2*n+2) * 23^(n - 1/2) * n^(3/2)). - Vaclav Kotesovec, Mar 18 2016 MATHEMATICA Table[Sum[Binomial[3*k, k]*Binomial[2*n-k, n]/(2*k+1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 18 2016 *) PROG (Maxima) taylor((sqrt(2)*sin(asin((3^(3/2)*sqrt(1-sqrt(1-4*x)))/2^(3/2))/3)* sqrt(1-sqrt(1-4*x)))/(sqrt(3)*(1-x/(1-(1-sqrt(1-4*x))/2)^2))/x, x, 0, 20); (PARI) for(n=0, 25, print1(sum(k=0, n, binomial(3*k, k)*binomial(2*n-k, n)/(2*k+1)), ", ")) \\ G. C. Greubel, Jun 05 2017 CROSSREFS Cf. A000108, A001764, A092392. Sequence in context: A177133 A186241 A193115 * A335819 A263853 A266083 Adjacent sequences:  A270486 A270487 A270488 * A270490 A270491 A270492 KEYWORD nonn AUTHOR Vladimir Kruchinin, Mar 18 2016 STATUS approved

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Last modified May 16 14:39 EDT 2021. Contains 343949 sequences. (Running on oeis4.)