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 A259554 a(n) = Sum_{i=0..n} (2^(i)*(-1)^(i+n)*C(n,i)*C(2*n+i-1,n-1)). 1
 1, 7, 52, 403, 3206, 25954, 212738, 1760035, 14666470, 122920642, 1035046816, 8749594462, 74207078908, 631140253072, 5381022869822, 45975731083555, 393556869530630, 3374504760608026, 28977403637496104, 249167023897718138 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 V. V. Kruchinin and D. V. Kruchinin, A Generating Function for the Diagonal T_{2n,n} in Triangles, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.6. FORMULA G.f.: A(x)=x*B(x)'/B(x), where B(x) is g.f. of A003168. Recurrence: 4*n*(2*n-1)*(17*n^2 - 51*n + 38)*a(n) = (1207*n^4 - 4828*n^3 + 6659*n^2 - 3662*n + 672)*a(n-1) - 2*(n-2)*(2*n-3)*(17*n^2 - 17*n + 4)*a(n-2). - Vaclav Kotesovec, Jul 01 2015 a(n) ~ (71+17*sqrt(17))^n / (17^(1/4) * sqrt(Pi*n) * 2^(4*n+1)). - Vaclav Kotesovec, Jul 01 2015 a(n) = 1/2*Sum_{k=0..n} binomial(n-1,n-k)*binomial(2*n+k-1,k). - Vladimir Kruchinin, Oct 07 2016 a(n) = n*hypergeom([2*n+1, -n+1], [2], -1). - Peter Luschny, Oct 07 2016 MAPLE a := n -> n*hypergeom([2*n+1, -n+1], [2], -1): seq(simplify(a(n)), n=1..9); # Peter Luschny, Oct 07 2016 MATHEMATICA Table[Sum[2^i * (-1)^(i+n) * Binomial[n, i] * Binomial[2*n+i-1, n-1], {i, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Jul 01 2015 *) PROG (Maxima) a(n):=sum(2^(i)*(-1)^(i+n)*binomial(n, i)*binomial(2*n+i-1, n-1), i, 0, n); (PARI) a(n) = sum(i=0, n, 2^i*(-1)^(i+n)*binomial(n, i)*binomial(2*n+i-1, n-1)); \\ Michel Marcus, Jul 02 2015 CROSSREFS Cf. A003168. Sequence in context: A246513 A015559 A097180 * A147962 A162233 A185623 Adjacent sequences:  A259551 A259552 A259553 * A259555 A259556 A259557 KEYWORD nonn AUTHOR Vladimir Kruchinin, Jun 30 2015 STATUS approved

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Last modified October 17 08:36 EDT 2019. Contains 328107 sequences. (Running on oeis4.)