A157125 A transform of the Catalan numbers.

1, -1, -1, 0, 2, 1, -1, -4, -2, 4, 12, 4, -20, -39, 3, 92, 118, -84, -388, -308, 596, 1528, 508, -3292, -5556, 1154, 16034, 17940, -18160, -71243, -45913, 127124, 290278, 46128, -710864, -1067564, 485108, 3504680, 3362756, -4957812, -15669148

Offset

0,5

Comments

Hankel transform is A157126. Partial sums are A157127.

Links

Table of n, a(n) for n=0..40.

Formula

G.f.: (1-x)*(sqrt(1+x^2+4*x^3)-sqrt(1+x^2))/(2*x^3*sqrt(1+x^2));

a(n) = Sum_{k=0..n} (-1)^binomial(n-k+1,2)*binomial(floor((n-k)/2),k)*A000108(k).

Conjecture: (n+3)*(n-2)*a(n) -4*a(n-1) +2*(n^2-n-4)*a(n-2) +2*(2*n^2-7*n+2)*a(n-3) +(n+1)*(n-4)*a(n-4) +2*(n-1)*(2*n-7)*a(n-5)=0. - R. J. Mathar, Nov 15 2012

Crossrefs

Cf. A000108, A157126, A157127.

Sequence in context: A123199 A212282 A346032 * A335941 A157143 A112096

Adjacent sequences: A157122 A157123 A157124 * A157126 A157127 A157128

Keyword

easy,sign

Author

Paul Barry, Feb 23 2009

Extensions

Divisor x^3 inserted in the g.f. - R. J. Mathar, Feb 06 2015

Status

approved