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A188314 Expansion of (1/(1-x))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108. 4
1, 2, 5, 16, 57, 219, 883, 3687, 15803, 69128, 307363, 1385003, 6310869, 29028616, 134610771, 628612921, 2953640371, 13953726888, 66240021987, 315812059436, 1511569447859, 7260364084997, 34984937594741, 169073568381936, 819288294835939, 3979892232651125, 19377475499900015 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
Hankel transform is the (25,-29) Somos-4 sequence A188315. Image of the Catalan numbers by A060098.
LINKS
FORMULA
G.f.: (1-x^2- sqrt(1-4*x-6*x^2+x^4))/(2*x).
G.f.: (1+x)/(1-x^2-x/(1-x-x/(1-x^2-x/(1-x-x/(1-...))))) (continued fraction).
a(n) = Sum{k=0..n, A000108(k)*Sum{i=0..floor(n/2), C(n-2i,n-2i-k)*C(k+i-1,i)}}.
Conjecture: (n+1)*a(n) +(n+2)*a(n-1) +(42-26*n)*a(n-2) +30*(3-n)*a(n-3) +(n-5)*a(n-4) +5*(n-6)*a(n-5)=0. - R. J. Mathar, Nov 15 2011
G.f. A(x) satisfies: A(x) = 1 + x * (1 + x*A(x) + A(x)^2). - Ilya Gutkovskiy, Jul 01 2020
MATHEMATICA
CoefficientList[Series[(1-x^2 - Sqrt[1-4*x-6*x^2+x^4])/(2*x), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)
PROG
(PARI) x='x+O('x^30); Vec((1-x^2- sqrt(1-4*x-6*x^2+x^4))/(2*x)) \\ G. C. Greubel, Aug 14 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2- Sqrt(1-4*x-6*x^2+x^4))/(2*x))); // G. C. Greubel, Aug 14 2018
CROSSREFS
Sequence in context: A072110 A323229 A197158 * A114296 A121689 A357580
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Mar 28 2011
STATUS
approved

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Last modified May 24 06:28 EDT 2024. Contains 372772 sequences. (Running on oeis4.)