A038806 Convolution of A008549 with A000302 (powers of 4).

0, 1, 10, 69, 406, 2186, 11124, 54445, 259006, 1205790, 5519020, 24918306, 111250140, 492051124, 2159081192, 9409526397, 40766269774, 175707380630, 753876367356, 3221460111958, 13716223138388, 58210889582796

Offset: 0

Author

Wolfdieter Lang

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, On the self-convolution of generalized Fibonacci numbers, arXiv:1703.00323 [math.CO], 2017.
• A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, J. Int. Seq. 10 (2007) # 07.9.7

Crossrefs

Cf. A000108, A000346, A008549, A000302, A002697.

Programs

Magma

[(n+3)*4^n -(n+2)*Binomial(2*n+3, n+1)/2: n in [0..25]]; // Vincenzo Librandi, Jun 09 2011

Mathematica

CoefficientList[Series[x ((1 - Sqrt[1 - 4 x])/(2 x)/(1 - 4 x))^2, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 29 2014 *)

Formula

a(n) = (n+3)*4^n -(n+2)*binomial(2*n+3, n+1)/2.

G.f.: x*(c(x)/(1-4*x))^2, where c(x) = g.f. for Catalan numbers A000108.

a(n+1), n >= 0 is convolution of A000346 with itself; a(n+1), n >= 0 is convolution of Catalan numbers A000108 C(n+1), n >= 0 with A002697; a(-1)=0.

Asymptotics: a(n) ~ 4^n*(n+1-4*sqrt(n/Pi)). - Fung Lam, Mar 28 2014

Recurrence: (n-1)*(n+1)*a(n) = 2*(n+1)*(4*n-3)*a(n-1) - 8*n*(2*n+1)*a(n-2). - Vaclav Kotesovec, Mar 28 2014