A070857 Expansion of (1+x*C)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.

1, 5, 19, 68, 240, 847, 3003, 10712, 38454, 138890, 504526, 1842392, 6760390, 24915555, 92196075, 342411120, 1275977670, 4769563590, 17879195130, 67197912600, 253172676120, 955992790038, 3617431679934, 13714878284368

Offset: 0

Author

N. J. A. Sloane, Jun 06 2002

Keywords

• nonn

Comments

If a zero is added in front, the sequence represents the Catalan transform of the squares A000290. [R. J. Mathar, Nov 06 2008]

a(n) is the number of North-East paths from (0,0) to (n+2,n+2) that cross y = x vertically exactly once and do not bounce off y = x to the right. Details can be found in Section 4.4 in Pan and Remmel's link. - Ran Pan, Feb 01 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Ran Pan, Jeffrey B. Remmel, Paired patterns in lattice paths, arXiv:1601.07988 [math.CO], 2016.

Programs

Mathematica

CoefficientList[Series[(1 + x (1 - (1 - 4 x)^(1/2)) / (2 x)) ((1 - (1 - 4 x)^(1/2)) / (2 x))^4, {x, 0,33}], x] (* Vincenzo Librandi, Apr 28 2017 *)

Maxima

a(n):=sum((k+1)^3*binomial(2*n-k,n),k,0,n)/(n+1); /* Vladimir Kruchinin, Apr 27 2017 */

PARI

C(x) = (1-(1-4*x)^(1/2))/(2*x);
x = 'x + O('x^30); Vec((1+x*C(x))*C(x)^4) \\ Michel Marcus, Feb 02 2016

Formula

a(n) = (Sum_{k=0..n} (k+1)^3*C(2*n-k,n))/(n+1). - Vladimir Kruchinin, Apr 27 2017

Conjecture: n*(n+4)*(13*n-1)*a(n) -2*(13*n+12)*(2*n+1)*(n+1)*a(n-1)=0. - R. J. Mathar, May 08 2017