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A157003
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Transform of Catalan numbers whose Hankel transform gives the Somos-4 sequence.
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1
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1, 1, 2, 4, 10, 27, 78, 234, 722, 2274, 7280, 23617, 77466, 256481, 856016, 2876940, 9728090, 33072228, 112974592, 387580856, 1334821448, 4613225722, 15994465796, 55615889745, 193904367362, 677709772035, 2374027931492, 8333765738127
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Image of the Catalan numbers A000108 by the Riordan array (1,x(1-x^2)). Hankel transform is A006720(n+2).
Partial sums of A157002.
Empirical: number of Dyck n-paths that avoid any one of {UDUDD, UUDDD, UUDUD, UUUDD}. e.g. of the 5 Dyck 3-paths UUDUDD contains UDUDD so a(3)=4. Also, number of Dyck n-paths that avoid DUD that ends at height of form 3*k+1, or that avoid UDU that ends at height of form 3*k-1. e.g. of the 5 Dyck 3-paths UUDUDD contains DUD ending at height 1 so a(3)=4. [From David Scambler, Mar 24 2011]
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FORMULA
| G.f.: c(x(1-x^2)) where c(x) is the g.f. of A000108;
a(n)=sum{k=0..n, (-1)^((n-k)/2)*(1+(-1)^(n-k))*C(k,floor((n-k)/2))*A000108(k)/2}.
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CROSSREFS
| Sequence in context: A104383 A205480 A108523 * A114507 A148105 A127386
Adjacent sequences: A157000 A157001 A157002 * A157004 A157005 A157006
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Feb 20 2009
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