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 A161797 G.f. satisfies: A(x) = 1/(1 - x/(1 - x*A(x))^3). 11
 1, 1, 4, 16, 71, 336, 1660, 8464, 44207, 235306, 1271807, 6961307, 38508659, 214950425, 1209170536, 6848080767, 39014400171, 223439516338, 1285660965508, 7428738358924, 43087099589998, 250766507928988, 1464026402082801 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 FORMULA a(n) = Sum_{k=0..n} C(n,k)/(n-k+1) * C(n+2*k-1,n-k). Let A(x)^m = Sum_{n>=0} a(n,m)*x^n then a(n,m) = Sum_{k=0..n} C(n+m-1,k)*m/(n-k+m) * C(n+2*k-1,n-k). G.f.: A(x) = (1/x)*serreverse[x/(1 + x/(1 - x)^3)]. Recurrence: 3*(n+1)*(3*n - 2)*(3*n + 2)*(2145*n^4 - 14355*n^3 + 33844*n^2 - 32668*n + 10380)*a(n) = 3*(115830*n^7 - 833085*n^6 + 2195691*n^5 - 2521863*n^4 + 998671*n^3 + 259048*n^2 - 263292*n + 41520)*a(n-1) + 3*(n-2)*(19305*n^6 - 129195*n^5 + 315651*n^4 - 367201*n^3 + 219176*n^2 - 66584*n + 7608)*a(n-2) + 3*(n-3)*(n-2)*(64350*n^5 - 334125*n^4 + 546005*n^3 - 255608*n^2 - 71320*n + 54328)*a(n-3) - 23*(n-4)*(n-3)*(n-2)*(2145*n^4 - 5775*n^3 + 3649*n^2 + 535*n - 654)*a(n-4). - Vaclav Kotesovec, Nov 18 2017 a(n) ~ sqrt(sqrt((159 + 100*sqrt(3))/13) - 2 - 5/sqrt(3)) * (3 + 2*sqrt(3) + sqrt(153 + 100*sqrt(3))/3)^(n+1) / (sqrt(Pi) * n^(3/2) * 2^(n + 5/2)). - Vaclav Kotesovec, Nov 18 2017 MATHEMATICA Table[Sum[Binomial[n, k]/(n-k+1) * Binomial[n+2*k-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Nov 18 2017 *) PROG (PARI) {a(n, m=1)=sum(k=0, n, binomial(n+m-1, k)*m/(n-k+m)*binomial(n+2*k-1, n-k))} CROSSREFS Cf. A109081, A321798, A321799. Sequence in context: A246014 A373959 A231373 * A124533 A227034 A224660 Adjacent sequences: A161794 A161795 A161796 * A161798 A161799 A161800 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 19 2009 STATUS approved

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Last modified August 4 17:39 EDT 2024. Contains 374923 sequences. (Running on oeis4.)