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Expansion of (1+x^3*C)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
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%I #6 May 30 2020 17:32:23

%S 1,1,2,6,16,47,146,471,1562,5291,18226,63648,224808,801686,2882452,

%T 10437745,38032110,139339635,512996370,1896907980,7041759120,

%U 26233530210,98046684060,367525880670,1381386710964,5205006015102

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

%H Michael De Vlieger, <a href="/A071726/b071726.txt">Table of n, a(n) for n = 0..1000</a>

%H Stoyan Dimitrov, <a href="https://arxiv.org/abs/2002.12322">On permutation patterns with constrained gap sizes</a>, arXiv:2002.12322 [math.CO], 2020.

%t With[{$MaxExtraPrecision = 1000}, CoefficientList[Series[# (1 + x^3*#) &[(1 - (1 - 4 x)^(1/2))/(2 x)], {x, 0, 25}], x]] (* _Michael De Vlieger_, May 30 2020 *)

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jun 06 2002