%I #51 Jan 14 2025 10:34:26

%S 0,1,42,594,4719,26026,111384,395352,1215126,3331251,8321170,19240650,

%T 41683005,85408596,166768096,312203232,563178924,982981701,1665911754,

%U 2749500754,4430505387,6985558206,10797503640,16388608600,24462014850,35952994935,52091785746

%N From enumerating paths in the plane.

%C a(n+1) is the determinant of the n X n Hankel matrix [C(i+j+3)]_{i,j=1..n} where C(n) = A000108(n), the n-th Catalan number. - _Michael Somos_, Jun 27 2023

%D R. P. Stanley, Enumerative Combinatorics, volume 1 (1986), p. 221, Example 4.5.18.

%H T. D. Noe, <a href="/A091962/b091962.txt">Table of n, a(n) for n = 0..1000</a>

%H Myriam de Sainte-Catherine, <a href="/A005700/a005700_1.pdf">Couplages et Pfaffiens en Combinatoire, Physique et Informatique</a>, PhD Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy of pages III.42-III.45)

%H G. Kreweras and H. Niederhausen, <a href="https://doi.org/10.1016/S0195-6698(81)80020-0">Solution of an enumerative problem connected with lattice paths</a>, European J. Combin., 2 (1981), 55-60.

%H Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 24.

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

%F a(n) = binomial(2*n+6, 7)*(2*n+3)*(n+1)*(n+2)/240.

%F G.f.: x*(1 + 31*x + 187*x^2 + 330*x^3 + 187*x^4 + 31*x^5 + x^6)/(1-x)^11. - _Colin Barker_, May 07 2012

%F a(n) = det(A*Transpose(A))/36, where A is the 2 X (n+1) matrix whose (i,j)-th element is j^(2*i-1). - _Lechoslaw Ratajczak_, Oct 01 2017

%F a(n) = binomial(2*n+4, 3)*binomial(2*n+6, 7)/160. - _G. C. Greubel_, Dec 17 2021

%F a(n) = a(-3-n) for all n in Z. - _Michael Somos_, Jun 27 2023

%F a(n) ~ n^10/4725. - _Stefano Spezia_, Dec 09 2023

%e G.f. = x + 42*x^2 + 594*x^3 + 4719*x^4 + 26026*x^5 + 111384*x^6 + ... - _Michael Somos_, Jun 27 2023

%t LinearRecurrence[{11,-55,165,-330,462,-462,330,-165,55,-11,1},{0,1,42,594,4719,26026,111384,395352,1215126,3331251,8321170},30] (* _Harvey P. Dale_, Apr 15 2017 *)

%o (PARI) a(n) = binomial(2*n+6, 7)*(2*n+3)*(n+1)*(n+2)/240; \\ _Michel Marcus_, Oct 13 2016

%o (Sage) [product(binomial(2*(n+j+2), 4*j+3) for j in (0..1))/160 for n in (0..30)] # _G. C. Greubel_, Dec 17 2021

%Y Cf. A000108.

%Y Cf. A000012, A000027, A000330, A006858 (Hankel determinants of Catalan numbers). - _Michael Somos_, Jun 27 2023

%K nonn,easy

%O 0,3

%A _Philippe Deléham_, Mar 13 2004