A302178 The number of 3D walks of semilength n in a quadrant returning to the origin.

1, 4, 40, 570, 9898, 195216, 4209084, 96941130, 2349133930, 59272544760, 1545550116240, 41416083787260, 1135679731004700, 31760915181412800, 903492759037272480, 26086451983000501410, 763124703525758894490, 22585374873810849150600, 675419388009799152812400

Offset

0,2

Links

Table of n, a(n) for n=0..18.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5. The sequence is type aab in Table 3.

Formula

a(n) = Sum_{i=0..n,j=0..n-i} A000108(i) * A000108(j) * A000984_(n-i-j) * (2n)!/((2i)!*(2j)!*(2n-2i-2j)!). - Nachum Dershowitz, Aug 13 2020

Conjecture D-finite with recurrence (n+1)*(n+2)^2*a(n) +2*(-29*n^3-39*n^2+8*n+6)*a(n-1) +36*(4*n-1)*(2*n-3)*(3*n-2)*a(n-2) -648*(2*n-5)*(n-2)*(2*n-3)*a(n-3)=0. - R. J. Mathar, Oct 29 2025

Maple

A302178 := proc(n)
     a :=0 ;
    for i from 0 to n do
        for j from 0 to n-i do
            a := a+A000108(i)*A000108(j)*A000984(n-i-j)*(2*n)!/(2*i)!/(2*j)!/(2*n-2*i-2*j)! ;
        end do:
    end do:
    a;
end proc:
seq(A302178(n), n=0..40) ; # R. J. Mathar, Oct 29 2025

Crossrefs

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Sequence in context: A290575 A196867 A276362 * A379252 A272261 A291817

Adjacent sequences: A302175 A302176 A302177 * A302179 A302180 A302181

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

a(8)-a(18) from Nachum Dershowitz, Aug 03 2020

Name edited by Nachum Dershowitz, Aug 13 2020

Status

approved