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A006320
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Royal paths in a lattice.
(Formerly M4200)
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5
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1, 6, 30, 146, 714, 3534, 17718, 89898, 461010, 2386390, 12455118, 65478978, 346448538, 1843520670, 9859734630, 52974158938, 285791932578, 1547585781414, 8408765223294
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973).
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
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FORMULA
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3-fold convolution of the large Schroeder numbers (A006318). G.f.=R^3, where R=[1-z-sqrt(1-6z+z^2)]/(2z) is the g.f. of A006318. - Emeric Deutsch, Mar 15 2004
a(n) = (3/n)*sum(binomial(n, j)*binomial(n+2+j, n-1), j=0..n) (n>0). - Emeric Deutsch, Aug 19 2004
Recurrence: (n+3)*(5*n-1)*a(n) = 2*(15*n^2+20*n+13)*a(n-1) - (5*n^2+5*n-24)*a(n-2) + (n-3)*a(n-3). - Vaclav Kotesovec, Oct 05 2012
a(n) ~ 3 * (1 + sqrt(2))^(2*n+3) / (2^(3/4) * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 05 2012, simplified Dec 24 2017
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MAPLE
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1, seq(3*sum(binomial(n, j)*binomial(n+2+j, n-1), j=0..n)/n, n=1..18);
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MATHEMATICA
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Table[SeriesCoefficient[(1-x-Sqrt[1-6*x+x^2])^3/(8*x^3), {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 05 2012 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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