A302181 Number of 3D walks of type abb.

1, 5, 62, 1065, 21714, 492366, 12004740, 308559537, 8255788970, 227976044010, 6457854821340, 186814834574550, 5500292590186380, 164387681345290500, 4976887208815547640, 152378485941172462785, 4711642301137121933850, 146964278352052950118770, 4619875954522866283392300

Offset

0,2

Comments

See Dershowitz (2017) for precise definition.

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.

Formula

a(n) = Sum_{k=0..2*n} binomial(2*n, k) * A005558(k) * A126869(2*n-k). - Mélika Tebni, Nov 06 2024

Conjecture D-finite with recurrence (n+1)^3*a(n) -2*(2*n-1)*(10*n^2+10*n+3)*a(n-1) +36*n*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 29 2025

Maple

C := n-> binomial(2*n, n)/(n+1): # Catalan numbers
A302181 := n-> add(binomial(2*n, k)*C(iquo(k+1, 2))*C(iquo(k, 2))*(2*iquo(k, 2)+1)*add((-1)^(k+j)*binomial(2*n-k, iquo(j, 2)), j=0..2*n-k), k=0..2*n): seq(A302181(n), n = 0 .. 18); # Mélika Tebni, Nov 06 2024

Crossrefs

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

Cf. A005558, A126869.

Sequence in context: A394654 A092823 A152031 * A357350 A360345 A371370

Adjacent sequences: A302178 A302179 A302180 * A302182 A302183 A302184

Keyword

nonn,walk

Author

N. J. A. Sloane, Apr 09 2018

Extensions

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

Status

approved