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 A064037 Number of walks of length 2n on cubic lattice, starting and finishing at origin and staying in first (nonnegative) octant. 13
 1, 3, 24, 285, 4242, 73206, 1403028, 29082339, 640672890, 14818136190, 356665411440, 8874875097270, 227135946200940, 5955171596514900, 159439898653636320, 4347741997166750235, 120493374240909299130, 3387806231071627372590, 96488484001399878973200 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS Nachum Dershowitz, Touchardâ€™s Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5. R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6. James Mallos, A 6-Letter 'DNA' for Baskets with Handles, Mathematics (2019) Vol. 7, No. 2, 165. G. Xin, Determinant formulas relating to tableaux of bounded height, Adv. Appl. Math. 45 (2010) 197-211. FORMULA 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) = binomial(2*n,n)*((7*n+11)*A002893(n+1)-(9*n+9)*A002893(n))/(2*(n+1)*(n+2)^2*(n+3)). - 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 EXAMPLE 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. MAPLE 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); series(ogf, x=0, 30); # Mark van Hoeij, Apr 19 2013 # 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: seq(a(n), n=0..20);  # Alois P. Heinz, Mar 29 2019 MATHEMATICA 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 *) PROG (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)); /* Joerg Arndt, Apr 19 2013 */ CROSSREFS Cf. A064036. The two- and one-dimensional equivalents are A005568 and A000108. Sequence in context: A081133 A218223 A276360 * A257453 A128572 A052592 Adjacent sequences:  A064034 A064035 A064036 * A064038 A064039 A064040 KEYWORD nonn AUTHOR Henry Bottomley, Aug 23 2001 EXTENSIONS Added more terms, Joerg Arndt, Apr 19 2013 STATUS approved

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Last modified August 7 12:19 EDT 2020. Contains 336276 sequences. (Running on oeis4.)