A217477 Z-sequence for the Riordan triangle A111125.

3, -4, 12, -40, 140, -504, 1848, -6864, 25740, -97240, 369512, -1410864, 5408312, -20801200, 80233200, -310235040, 1202160780, -4667212440, 18150270600, -70690527600, 275693057640, -1076515748880, 4208197927440, -16466861455200

Offset: 0

Author

Wolfdieter Lang, Oct 18 2012

Keywords

• sign
• easy

Comments

For the notion Z-sequence for a Riordan triangle (lower triangular matrix) R(n,m) see a W.Lang link under A006232, with references. The Z-sequence appears in the recurrence for any entry R(n,0), n >= 1: R(n,0) = sum(Z(m)*R(n-1,m), m=0..n-1).

The A-sequence for the Riordan triangle A111125 is (-1)^n*A115141(n).

Crossrefs

Cf.: A111125, A115141.

Formula

O.g.f.: (1 - (2 - c(-x))/(1 + 4*x))/(1 - c(-x)) = ((3 + 4*x) + 4*x*c(-x))/(1 + 4*x), with c(x) the o.g.f. of A000108 (Catalan).

a(0) = 3, a(n) = ((-1)^n)*4*binomial(2*n-1,n), n >= 1, (from adding the two pieces of the second o.g.f. version).

|a(n)| = A100320(n), n >0. - R. J. Mathar, Apr 22 2013

G.f.: 1 + G(0), where G(k)= 1 + 1/(1 - 2*x*(2*k+1)/(2*x*(2*k+1) - (k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013