A068764 Generalized Catalan numbers.

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

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