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A064037
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Number of walks of length 2n on cubic lattice, starting and finishing at origin and staying in first (nonnegative) octant.
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13
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1, 3, 24, 285, 4242, 73206, 1403028, 29082339, 640672890, 14818136190, 356665411440, 8874875097270, 227135946200940, 5955171596514900, 159439898653636320, 4347741997166750235, 120493374240909299130, 3387806231071627372590, 96488484001399878973200
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{j=0..n} C(2n, 2j)*c(j)*c(j+1)*c(n-j) where c(k)=A000108(k).
G.f. is a large expression in terms of hypergeometric functions and sqrt's, see Maple program. - Mark van Hoeij, Apr 19 2013
a(n) ~ 2^(2*n - 2) * 3^(2*n + 9/2) / (Pi^(3/2) * n^(9/2)). - Vaclav Kotesovec, Jun 09 2019
D-finite with recurrence: (n+3)*(n+2)*(n+1)*a(n) -4*(2*n-1)*(5*n^2+10*n+3)*a(n-1) +36*(n-1)*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Feb 20 2020
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EXAMPLE
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a(1)=3 and a(2)=24 since if the possible steps are Right, Left, Up, Down, Forwards and Backwards, then the two-step paths are FB, RL and UD, while the four-step paths are FBFB, FBRL, FBUD, FFBB, FRBL, FRLB, FUBD, FUDB, RFBL, RFLB, RLFB, RLRL, RLUD, RRLL, RUDL, RULD, UDFB, UDRL, UDUD, UFBD, UFDB, URDL, URLD, UUDD.
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MAPLE
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f := -3*x+(1+sqrt(1-40*x+144*x^2))/4;
H := (1-2*f)*f*hypergeom([1/6, 1/3], [1], 27*(1-2*f)*f^2)^2/sqrt(1+6*f);
r2 := (1-4*x)*(36*x-1)*(1920*x^2+166*x+1)*x^2;
r1 := -(138240*x^4+7776*x^3+200*x^2-92*x-1)*x;
r0 := 19800*x^3+764*x^2-86*x-1;
ogf := (r2*diff(H, x, x)+r1*diff(H, x)+r0*H)/(5760*x^4) + 1/(2*x);
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 2*n+1, ((8*n-4)*(5*n^2+10*n+3)
*a(n-1)-36*(2*n-1)*(2*n-3)*(n-1)*a(n-2))/((n+1)*(n+2)*(n+3)))
end:
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MATHEMATICA
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Table[Sum[Binomial[2*n, 2*j] * CatalanNumber[j] * CatalanNumber[j+1] * CatalanNumber[n-j], {j, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jun 09 2019 *)
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PROG
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(PARI)
C(n, k) = binomial(n, k);
c(n) = binomial(2*n, n)/(n+1);
a(n) = sum(j=0, n, C(2*n, 2*j)*c(j)*c(j+1)*c(n-j));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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