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A071725
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Expansion of (1+x^2*C^4)*C, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
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1
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1, 1, 3, 10, 34, 117, 407, 1430, 5070, 18122, 65246, 236436, 861764, 3157325, 11622015, 42961470, 159419670, 593636670, 2217608250, 8308432140, 31212003420, 117544456770, 443690433654, 1678353186780, 6361322162444
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)=number of Dyck (n+3)-paths for which the first downstep followed by an upstep (or by nothing at all) is in position 6. For example, a(2)=3 counts UUUUDdUDDD, UUUDDdUUDD, UUUDDdUDUD (the downstep in position 6 is in small type). - David Callan (callan(AT)stat.wisc.edu), Dec 09 2004
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FORMULA
| Contribution from Paul Barry (pbarry(AT)wit.ie), Jun 28 2009: (Start)
E.g.f.: exp(2x)*dif(Bessel_I(1,2x)-Bessel_I(2,2x),x);
a(n)=sum{k=0..n, C(n,k)*2^(n-k)*(-1)^k*C(k+1,floor(k/2))}. (End)
Conjecture: (n+31)*(n+3)*a(n) +(n^2-180*n-219)*a(n-1) -10*(2*n-3)*(n-10)*a(n-2) = 0 . - R. J. Mathar, Nov 23 2011
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CROSSREFS
| Essentially the same as A026016.
Sequence in context: A048580 A059738 A094832 * A026016 A109263 A136439
Adjacent sequences: A071722 A071723 A071724 * A071726 A071727 A071728
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Jun 06 2002
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