A302179 The number of 3D walks of length n in an octant returning to axis of origin.

1, 1, 4, 9, 40, 120, 570, 1995, 9898, 38178, 195216, 805266, 4209084, 18239364, 96941130, 436235085, 2349133930, 10891439130, 59272544760, 281544587610, 1545550116240, 7489973640240, 41416083787260, 204122127237210, 1135679731004700, 5678398655023500, 31760915181412800, 160789633105902300

Offset

0,3

Links

Table of n, a(n) for n=0..27.

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

Mélika Tebni, Fonctions de Bessel et cheminements en 3D, Dec 2024.

Formula

a(n) = Sum_{i=0..n, j=0..n-i, i,j even} A126120(i) * A126120(j) * A001405(n-i-j) * n!/(i! * j! * (n-i-j)!). - Nachum Dershowitz, Aug 06 2020

E.g.f.: (BesselI(1, 2*x)/x)^2*(BesselI(0, 2*x) + BesselI(1, 2*x)). - Mélika Tebni, Jan 06 2025

Prog

(PARI) C(n) = binomial(2*n, n)/(n+1); \\ A000108
f(n) = binomial(n, floor(n/2)); \\ A001405
a(n) = sum(i=0, n, if (!(i%2), sum(j=0, n-i, if (!(j%2), C(i/2)*C(j/2)*f(n-i-j)*n!/(i! * j! * (n-i-j)!))))); \\ Michel Marcus, Aug 07 2020

Crossrefs

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

Sequence in context: A149163 A149164 A238420 * A370603 A370602 A354738

Adjacent sequences: A302176 A302177 A302178 * A302180 A302181 A302182

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

a(13)-a(27) from Nachum Dershowitz, Aug 04 2020

Status

approved