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

0,2

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.

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

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 *)

Prog

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

Crossrefs

Sequence in context: A001435 A359293 A092492 * A143954 A047145 A240525

Adjacent sequences: A070854 A070855 A070856 * A070858 A070859 A070860

Keyword

nonn

Author

N. J. A. Sloane, Jun 06 2002

Status

approved