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

0,3

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

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A026958 A026988 A081280 * A016273 A128737 A320228

Adjacent sequences: A038803 A038804 A038805 * A038807 A038808 A038809

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved