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A080893
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E.g.f. exp(x*C(x)) = exp((1-sqrt(1-4*x))/2), where C(x) is the g.f. of the Catalan numbers A000108.
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4
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1, 1, 3, 19, 193, 2721, 49171, 1084483, 28245729, 848456353, 28875761731, 1098127402131, 46150226651233, 2124008553358849, 106246577894593683, 5739439214861417731, 332993721039856822081, 20651350143685984386753
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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E.g.f.: exp((1-sqrt(1-4*x))/2).
D-finite with recurrence: a(n+2) = 2( 2 n + 1 ) a(n+1) + a(n)
Recurrence: y(n+1) = sum(k=0..n, binomial(n, k)*binomial(2k, k)*k!*y(n-k) ).
G.f.: 1 + x/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013
a(n) = hypergeom([-n+1, n], [], -1). - Peter Luschny, Oct 17 2014
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MATHEMATICA
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y[x_] := y[x] = 2(2x - 3)y[x - 1] + y[x - 2]; y[0] = 1; y[1] = 1; Table[y[n], {n, 0, 17}]
With[{nn=20}, CoefficientList[Series[Exp[(1-Sqrt[1-4x])/2], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Oct 30 2011 *)
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PROG
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(PARI) {a(n) = if( n<1, n = 1 - n); n! * polcoeff( exp( (1 - sqrt(1 - 4*x + x * O(x^n))) / 2), n)} /* Michael Somos, Apr 07 2012 */
(Sage)
A080893 = lambda n: hypergeometric([-n+1, n], [], -1)
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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STATUS
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approved
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