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%I #25 Sep 08 2022 08:44:53
%S 0,1,10,69,406,2186,11124,54445,259006,1205790,5519020,24918306,
%T 111250140,492051124,2159081192,9409526397,40766269774,175707380630,
%U 753876367356,3221460111958,13716223138388,58210889582796
%N Convolution of A008549 with A000302 (powers of 4).
%H Vincenzo Librandi, <a href="/A038806/b038806.txt">Table of n, a(n) for n = 0..200</a>
%H Hacène Belbachir, Toufik Djellal, Jean-Gabriel Luque, <a href="https://arxiv.org/abs/1703.00323">On the self-convolution of generalized Fibonacci numbers</a>, arXiv:1703.00323 [math.CO], 2017.
%H A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Rinaldi/rinaldi5.html">Permutations defining convex permutominoes</a>, J. Int. Seq. 10 (2007) # 07.9.7
%F a(n) = (n+3)*4^n -(n+2)*binomial(2*n+3, n+1)/2.
%F G.f.: x*(c(x)/(1-4*x))^2, where c(x) = g.f. for Catalan numbers A000108.
%F 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.
%F Asymptotics: a(n) ~ 4^n*(n+1-4*sqrt(n/Pi)). - _Fung Lam_, Mar 28 2014
%F 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
%t CoefficientList[Series[x ((1 - Sqrt[1 - 4 x])/(2 x)/(1 - 4 x))^2, {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 29 2014 *)
%o (Magma) [(n+3)*4^n -(n+2)*Binomial(2*n+3, n+1)/2: n in [0..25]]; // _Vincenzo Librandi_, Jun 09 2011
%Y Cf. A000108, A000346, A008549, A000302, A002697.
%K easy,nonn
%O 0,3
%A _Wolfdieter Lang_
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