A091962 From enumerating paths in the plane.

0, 1, 42, 594, 4719, 26026, 111384, 395352, 1215126, 3331251, 8321170, 19240650, 41683005, 85408596, 166768096, 312203232, 563178924, 982981701, 1665911754, 2749500754, 4430505387, 6985558206, 10797503640, 16388608600, 24462014850, 35952994935, 52091785746

Offset: 0

Author

Philippe Deléham, Mar 13 2004

Keywords

• nonn
• easy

Comments

a(n+1) is the determinant of the n X n Hankel matrix [C(i+j+3)]A000108(n),%20the%20n-th%20Catalan%20number.%20-%20_Michael%20Somos">{i,j=1..n} where C(n) = A000108(n), the n-th Catalan number. - _Michael Somos, Jun 27 2023

Examples

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

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Myriam de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire, Physique et Informatique, PhD Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy of pages III.42-III.45)
• G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.
• Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See p. 24.
• Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Crossrefs

Cf. A000108.

Cf. A000012, A000027, A000330, A006858 (Hankel determinants of Catalan numbers). - Michael Somos, Jun 27 2023

Programs

Mathematica

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 *)

PARI

a(n) = binomial(2*n+6, 7)*(2*n+3)*(n+1)*(n+2)/240; \\ Michel Marcus, Oct 13 2016

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

Formula

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

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

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

a(n) = binomial(2*n+4, 3)*binomial(2*n+6, 7)/160. - G. C. Greubel, Dec 17 2021

a(n) = a(-3-n) for all n in Z. - Michael Somos, Jun 27 2023

a(n) ~ n^10/4725. - Stefano Spezia, Dec 09 2023