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 A068764 Generalized Catalan numbers. 15
 1, 1, 4, 18, 88, 456, 2464, 13736, 78432, 456416, 2697088, 16141120, 97632000, 595912960, 3665728512, 22703097472, 141448381952, 885934151168, 5575020435456, 35230798994432, 223485795258368, 1422572226146304, 9083682419818496, 58169612565614592, 373486362257899520, 2403850703479816192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS a(n) = K(2,2; n)/2 with K(a,b; n) defined in a comment to A068763. Hankel transform is A166232(n+1). - Paul Barry, Oct 09 2009 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA G.f.: (1-sqrt(1-8*x*(1-x)))/(4*x). a(n+1) = 2*sum(a(k)*a(n-k), k=0..n), n>=1, a(0) = 1 = a(1). a(n) = (2^n)*p(n, -1/2) with the row polynomials p(n, x) defined from array A068763. E.g.f. (offset -1) is exp(4*x)*BesselI(1, 2*sqrt(2)*x)/(sqrt(2)*x). - Vladeta Jovovic, Mar 31 2004 The o.g.f. satisfies A(x) = 1 + x*(2*A(x)^2 - 1), A(0) = 1. - Wolfdieter Lang, Nov 13 2007 a(n) = subs(t=1,(d^(n-1)/dt^(n-1))(-1+2*t^2)^n)/n!, n >= 2, due to the Lagrange series for the given implicit o.g.f. equation. This formula holds also for n=1 if no differentiation is used. - Wolfdieter Lang, Nov 13 2007, Feb 22 2008 1/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-x/(1-x-2x/(1-..... (continued fraction). - Paul Barry, Jan 29 2009 a(n) = A166229(n)/(2-0^n). - Paul Barry, Oct 09 2009 a(n) = sum(binomial(n-1,k-1)*1/k*sum(binomial(k,j)*binomial(k+j,j-1),j,1,k),k,1,n), n>0. - Vladimir Kruchinin, Aug 11 2010 D-finite with recurrence: (n+1)*a(n) = 4*(2*n-1)*a(n-1) - 8*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 13 2012 a(n) ~ sqrt(1+sqrt(2))*(4+2*sqrt(2))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 13 2012 a(n) = 4^(n-1)*hypergeom([(1-n)/2,1-n/2], [2], 1/2) + 0^n/sqrt(2). - Vladimir Reshetnikov, Nov 07 2015 0 = a(n)*(+64*a(n+1) - 160*a(n+2) + 32*a(n+3)) + a(n+1)*(+32*a(n+1) + 48*a(n+2) - 20*a(n+3)) + a(n+2)*(+4*a(n+2) + a(n+3)) for all n>=0. - Michael Somos, Nov 08 2015 a(n) = (-1)^n * Sum_{k=0..n} (-2)^k * binomial(n,k) * binomial(2*k+1,n) / (2*k+1). - Seiichi Manyama, Jul 24 2023 EXAMPLE G.f. = 1 + x + 4*x^2 + 18*x^3 + 88*x^4 + 456*x^5 + 2464*x^6 + 13736*x^7 + ... MATHEMATICA Table[SeriesCoefficient[(1-Sqrt[1-8*x*(1-x)])/(4*x), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2012 *) Round@Table[4^(n-1) Hypergeometric2F1[(1-n)/2, 1-n/2, 2, 1/2] + KroneckerDelta[n]/Sqrt[2], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 07 2015 *) a[ n_] := If[ n < 1, Boole[n == 0], 4^(n - 1) Hypergeometric2F1[ (1 - n)/2, (2 - n)/2, 2, 1/2]]; (* Michael Somos, Nov 08 2015 *) PROG (PARI) {a(n) = my(A); if( n<1, n==0, n--; A = x * O(x^n); n! * simplify( polcoeff( exp(4*x + A) * besseli(1, 2*x * quadgen(8) + A), n)))}; /* Michael Somos, Mar 31 2007 */ (Maxima) a(n):=sum(binomial(n-1, k-1)*1/k*sum(binomial(k, j)*binomial(k+j, j-1), j, 1, k), k, 1, n); \\ Vladimir Kruchinin, Aug 11 2010 (PARI) x='x+O('x^66); Vec((1-sqrt(1-8*x*(1-x)))/(4*x)) \\ Joerg Arndt, May 06 2013 CROSSREFS Cf. A025227, A025228, A025229, A025230. Cf. A071356, A001003, A025235. Sequence in context: A244785 A260650 A006629 * A127394 A046984 A129323 Adjacent sequences: A068761 A068762 A068763 * A068765 A068766 A068767 KEYWORD nonn,easy AUTHOR Wolfdieter Lang, Mar 04 2002 STATUS approved

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Last modified September 18 01:15 EDT 2024. Contains 375995 sequences. (Running on oeis4.)