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A091962 From enumerating paths in the plane. 8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
REFERENCES
R. P. Stanley, Enumerative Combinatorics, volume 1 (1986), p. 221, Example 4.5.18.
LINKS
M. de Sainte-Catherine, Couplages et Pfaffiens en Combinatoire. Physique et Informatique, Ph.D Dissertation, Université Bordeaux I, 1983. (Annotated scanned copy)
G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
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
EXAMPLE
G.f. = x + 42*x^2 + 594*x^3 + 4719*x^4 + 26026*x^5 + 111384*x^6 + ... - Michael Somos, Jun 27 2023
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 *)
PROG
(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
CROSSREFS
Cf. A000108.
Cf. A000012, A000027, A000330, A006858 (Hankel determinants of Catalan numbers). - Michael Somos, Jun 27 2023
Sequence in context: A293096 A279888 A104901 * A269659 A007746 A200853
KEYWORD
nonn,easy
AUTHOR
Philippe Deléham, Mar 13 2004
STATUS
approved

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Last modified March 29 04:59 EDT 2024. Contains 371264 sequences. (Running on oeis4.)