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 A100193 a(n) = Sum_{k=0..n} binomial(2n,n+k)*3^k. 2
 1, 5, 27, 146, 787, 4230, 22686, 121476, 649731, 3472382, 18546922, 99023292, 528535726, 2820451964, 15048601308, 80283276936, 428271193827, 2284478396334, 12185310873138, 64993897108236, 346655914156602 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A transform of 3^n under the mapping g(x)->(1/sqrt(1-4x))g(x*c(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. A transform of 4^n under the mapping g(x)->(1/(c(x)*sqrt(1-4x))g(x*c(x)). Hankel transform is A127357. In general, the Hankel transform of Sum_{k=0..n} C(2n,k)*r^(n-k) is the sequence with g.f. 1/(1-2x+r^2*x^2). - Paul Barry, Jan 11 2007 LINKS FORMULA G.f.: (sqrt(1-4x)+1)/(sqrt(1-4x)*(4*sqrt(1-4x)-2)). G.f.: sqrt(1-4x)*(3*sqrt(1-4x)-8x+3)/((1-4x)(6-32x)). a(n) = Sum_{k=0..n} binomial(2n, n-k)*3^k. a(n) = (Sum_{k=0..n} binomial(2n, n-k))*(Sum_{j=0..n} binomial(n, j)*(-1)^(n-j)*4^j). a(n) = Sum_{k=0..n} C(2n,k)*3^(n-k). - Paul Barry, Jan 11 2007 a(n) = Sum_{k=0..n} C(n+k-1,k)*4^(n-k). - Paul Barry, Sep 28 2007 Conjecture: 9*n*a(n) + 6*(11-18*n)*a(n-1) + 16*(26*n-37)*a(n-2) + 256*(5-2*n)*a(n-3) = 0. - R. J. Mathar, Nov 09 2012 a(n) ~ (16/3)^n. - Vaclav Kotesovec, Feb 03 2014 a(n) = [x^n] 1/((1 - x)^n*(1 - 4*x)). - Ilya Gutkovskiy, Oct 12 2017 MATHEMATICA Table[Binomial[2*n, n]*Hypergeometric2F1[1, -n, 1+n, -3], {n, 0, 20}] (* Vaclav Kotesovec, Feb 03 2014 *) CROSSREFS Cf. A032443, A100192, A000108, A127357. Sequence in context: A293295 A015535 A026292 * A158869 A162557 A134425 Adjacent sequences:  A100190 A100191 A100192 * A100194 A100195 A100196 KEYWORD easy,nonn AUTHOR Paul Barry, Nov 08 2004 STATUS approved

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Last modified May 26 16:52 EDT 2020. Contains 334627 sequences. (Running on oeis4.)