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

0,3

Comments

Also equals row sums of triangle A125080.

Links

Table of n, a(n) for n=0..21.

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

Example

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.

Prog

(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 )}

Crossrefs

Cf. A115080, A115082 (column 1), A115083 (column 2), A115084 (row sums); A115086.

Cf. A125080 (related triangle); A000108, A001147.

Sequence in context: A024333 A024334 A162477 * A323672 A103466 A346762

Adjacent sequences: A115078 A115079 A115080 * A115082 A115083 A115084

Keyword

nonn

Author

Paul D. Hanna, Jan 13 2006, Nov 19 2006

Status

approved