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A098658
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a(n) = 3^n*(2*n)!/(n!)^2.
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7
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1, 6, 54, 540, 5670, 61236, 673596, 7505784, 84440070, 956987460, 10909657044, 124965162504, 1437099368796, 16581915793800, 191876454185400, 2225766868550640, 25874539846901190, 301362287628613860
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OFFSET
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0,2
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COMMENTS
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Number of lattice paths from (0,0) to (n,n) using steps (0,1) and three kinds of steps (1,0). - Joerg Arndt, Jul 01 2011
Sixth binomial transform of 1/sqrt(1-36*x^2).
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LINKS
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FORMULA
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G.f.: 1/sqrt((1-6*x)^2-36*x^2) = 1/sqrt(1-12*x).
E.g.f.: exp(6*x)*BesselI(0, 6x).
a(n) = [t^n](1+6*t+9*t^2)^n.
G.f.: Q(0), where Q(k) = 1 + 12*x*(4*k+1)/( 4*k+2 - 12*x*(4*k+2)*(4*k+3)/(12*x*(4*k+3) + 4*(k+1)/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013
exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + 6*x + 45*x^2 + 378*x^3 + ... is the o.g.f. for A101600. - Peter Bala, Jul 16 2015
Sum_{n>=0} 1/a(n) = 12/11 + 12*sqrt(11)*arcsin(1/sqrt(12))/121.
Sum_{n>=0} (-1)^n/a(n) = 12/13 - 12*sqrt(13)*arcsinh(1/sqrt(12))/169. (End)
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MATHEMATICA
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PROG
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(PARI) /* same as in A092566 but use */
steps=[[1, 0], [1, 0], [1, 0], [0, 1]]; /* note the triple [1, 0] */
(Magma) [3^n*Factorial(2*n)/Factorial(n)^2: n in [0..20]]; // Vincenzo Librandi, Jul 05 2011
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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