login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

From enumerating paths in the plane.
8

%I #48 Sep 20 2024 04:15:37

%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 <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