A071748 Expansion of (1+x^4*C^2)*C^3, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 3, 9, 28, 91, 302, 1021, 3507, 12209, 42991, 152866, 548148, 1979990, 7198055, 26316577, 96701545, 356941485, 1322891115, 4920919230, 18366138480, 68756266170, 258118218960, 971492416674, 3665120443878, 13857614540714, 52501491416982, 199285818849476, 757787703002008

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..1665

Formula

(-6+4*n)*a(n) + (-6-13*n)*a(n+1) + (16+7*n)*a(n+2) + (-22-5*n)*a(n+3) + (32+5*n)*a(n+4) + (-8-n)*a(n+5) = 0. - Robert Israel, Jul 30 2018

a(n) = binomial(2*n,n)*(197*n^4-96*n^3-401*n^2+24*n+252)/(4*(n+1)*(n+2)*(n+3)*(2*n-3)*(2*n-1)) for n > 1. - Tani Akinari, Aug 05 2025

a(n) ~ 197 * 4^(n-2) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Oct 04 2025

Maple

f:= gfun:-rectoproc({(-6+4*n)*a(n)+(-6-13*n)*a(n+1)+(16+7*n)*a(n+2)+(-22-5*n)*a(n+3)+(32+5*n)*a(n+4)+(-8-n)*a(n+5), a(0) = 1, a(1) = 3, a(2) = 9, a(3) = 28, a(4) = 91, a(5) = 302, a(6) = 1021}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Jul 30 2018

Mathematica

Join[{1, 3}, RecurrenceTable[{(-13n-6) a[n+1] + (7n+16) a[n+2] + (-5n-22)* a[n+3] + (5n+32) a[n+4] + (-n-8) a[n+5] + (4n-6) a[n] == 0, a[2] == 9, a[3] == 28, a[4] == 91, a[5] == 302, a[6] == 1021}, a, {n, 2, 30}]] (* Jean-François Alcover, Aug 29 2022, after Robert Israel *)

Prog

(PARI) C = (1-(1-4*x)^(1/2))/(2*x); Vec((1+x^4*C^2)*C^3 + O(x^20)) \\ Felix Fröhlich, Jul 30 2018
(Maxima) a(n):=if n<2 then 3^n else binomial(2*n, n)*(197*n^4-96*n^3-401*n^2+24*n+252)/(4*(n+1)*(n+2)*(n+3)*(2*n-3)*(2*n-1));
makelist(a(n), n, 0, 50); /* Tani Akinari, Aug 05 2025 */

Crossrefs

Cf. A000108.

Sequence in context: A189940 A047047 A071744 * A071752 A071756 A176673

Adjacent sequences: A071745 A071746 A071747 * A071749 A071750 A071751

Keyword

nonn,easy

Author

N. J. A. Sloane, Jun 06 2002

Status

approved