A161006 Convolution of A000108 (Catalan numbers) with A126120 (Catalan numbers interpolated with 0's).

1, 1, 3, 6, 18, 49, 155, 486, 1614, 5414, 18630, 64828, 228740, 814485, 2926323, 10588486, 38561814, 141214570, 519711666, 1921126036, 7129756188, 26555090618, 99228108222, 371886366620, 1397548389644, 5265130603468

Offset: 0

Author

Philippe Deléham, Jun 02 2009

Keywords

• nonn

Crossrefs

Cf. A000108, A126120.

Programs

Maple

C := x-> (1/2-(1/2)*sqrt(1-4*x))/x: G := C(x)*C(x^2): Gser := series(G, x = 0, 30): seq(coeff(Gser, x, n), n = 0 .. 27); # Emeric Deutsch, Jun 22 2009

Formula

G.f.: C(x)C(x^2), where C(x) = (1-sqrt(1-4x))/(2x) is the Catalan function. - Emeric Deutsch, Jun 22 2009

Conjecture: -(47*n-42)*(n+3)*(n+2)*(n+1)*a(n) + 2*(n+2)*(n+1)*(270*n^2 - 523*n + 126)*a(n-1) - 8*(n+1)*(211*n^3 - 1022*n^2 + 946*n + 63)*a(n-2) + 8*(-212*n^4 - 326*n^3 + 2489*n^2 - 1636*n + 126)*a(n-3) + 16*(985*n^4 - 8154*n^3 + 23771*n^2 - 28164*n + 11151)*a(n-4) + 32*(-386*n^4 + 5771*n^3 - 29314*n^2 + 60721*n - 43533)*a(n-5) - 64*(n-4)*(2*n-9)*(258*n^2 - 1193*n + 1267)*a(n-6) + 128*(n-5)*(2*n-9)*(2*n-11)*(82*n-203)*a(n-7) = 0. - R. J. Mathar, Oct 04 2014

Extensions

Extended by Emeric Deutsch, Jun 22 2009