A115081 Column 0 of triangle A115080.
1, 1, 3, 11, 50, 257, 1467, 9081, 60272, 424514, 3151226, 24510411, 198870388, 1676878231, 14648843341, 132228263355, 1230505582380, 11782173683640, 115878367974480, 1168833058344870, 12075008262774120, 127608480923659770
Offset: 0
Keywords
Examples
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.
Crossrefs
Programs
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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 )}
Formula
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, Feb 19 2007
Comments