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A091962
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From enumerating paths in the plane.
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8
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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
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OFFSET
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0,3
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COMMENTS
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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
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, volume 1 (1986), p. 221, Example 4.5.18.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
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FORMULA
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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
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EXAMPLE
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G.f. = x + 42*x^2 + 594*x^3 + 4719*x^4 + 26026*x^5 + 111384*x^6 + ... - Michael Somos, Jun 27 2023
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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