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URL: https://oeis.org/A070857

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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.
6
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
Ran Pan and 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
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
a(n) = 2*A002057(n) - A000245(n+1) = (13*n+12) * (2*n+2)!/(n! * (n+4)!). - Seiichi Manyama, Nov 25 2025
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);
my(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
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jun 06 2002
STATUS
approved