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%I #21 May 08 2017 13:03:25
%S 1,5,19,68,240,847,3003,10712,38454,138890,504526,1842392,6760390,
%T 24915555,92196075,342411120,1275977670,4769563590,17879195130,
%U 67197912600,253172676120,955992790038,3617431679934,13714878284368
%N Expansion of (1+x*C)*C^4, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
%C If a zero is added in front, the sequence represents the Catalan transform of the squares A000290. [_R. J. Mathar_, Nov 06 2008]
%C 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
%H Vincenzo Librandi, <a href="/A070857/b070857.txt">Table of n, a(n) for n = 0..1000</a>
%H Ran Pan, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%F a(n) = (Sum_{k=0..n} (k+1)^3*C(2*n-k,n))/(n+1). - _Vladimir Kruchinin_, Apr 27 2017
%F 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
%t 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 *)
%o (PARI) C(x) = (1-(1-4*x)^(1/2))/(2*x);
%o x = 'x + O('x^30); Vec((1+x*C(x))*C(x)^4) \\ _Michel Marcus_, Feb 02 2016
%o (Maxima)
%o a(n):=sum((k+1)^3*binomial(2*n-k,n),k,0,n)/(n+1); /* _Vladimir Kruchinin_, Apr 27 2017 */
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jun 06 2002
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