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Search: id:A115081
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| 1, 1, 3, 11, 50, 257, 1467, 9081, 60272, 424514, 3151226, 24510411, 198870388, 1676878231, 14648843341, 132228263355, 1230505582380, 11782173683640, 115878367974480, 1168833058344870, 12075008262774120, 127608480923659770
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OFFSET
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0,3
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COMMENT
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Also equals row sums of triangle A125080.
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FORMULA
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a(n) = Sum_{k=0..[n/2]} A000108(n-k)*A001147(k)*C(n,2*k), where A000108 is the Catalan numbers and A001147 is the double factorials.
a(n) = Sum_{k=0..[n/2]} A000108(n-k)*A000108(k)*(k+1)!*C(n,2k)/2^k where A000108(n) = C(2n,n)/(n+1) are the Catalan numbers. a(n) = Sum_{k=0..n} (-1)^(n-k)*n!/k!*A115082(k) . - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 19 2007
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EXAMPLE
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At n=5, a(5) = Sum_{k=0..2} A000108(5-k)*A001147(k)*C(5,2*k) so that a(5) = 42*1*C(5,0) + 14*1*C(5,2) + 5*3*C(5,4) = 42*1*1 + 14*1*10 + 5*3*5 = 42 + 140 + 75 = 257.
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PROGRAM
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(PARI) {a(n)=sum(k=0, n\2, binomial(2*n-2*k, n-k)/(n-k+1)*binomial(2*k, k)*k!/2^k*binomial(n, 2*k))}
(PARI) {a(n)=sum(k=0, n\2, (2*n-2*k)!*n!/k!/(n-k)!/(n-k+1)!/(n-2*k)!/2^k )}
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CROSSREFS
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Cf. A115080, A115082 (column 1), A115083 (column 2), A115084 (row sums); A115086.
Cf. A125080 (related triangle); A000108, A001147.
Sequence in context: A024333 A024334 A162477 this_sequence A103466 A000254 A081048
Adjacent sequences: A115078 A115079 A115080 this_sequence A115082 A115083 A115084
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jan 13 2006, Nov 19 2006
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