A049037 Number of cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.

1, 6, 54, 996, 22734, 577692, 15680628, 445162392, 13055851998, 392475442092, 12029082873372, 374482032292008, 11808861461931492, 376406128925067528, 12108063535794336312, 392560994063887113744, 12814685828476778001726, 420836267423433182275404

Offset: 0

Author

Alessandro Zinani (alzinani(AT)tin.it)

Keywords

• easy
• nonn
• nice

Examples

			a(5) = 577692 because there are 577692 different walks that start and end at the origin after 2*5=10 steps, avoiding origin at intermediate steps.
		

References

• S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200
• Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice
• Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Cached copy, with permission of the author]
• N. J. A. Sloane, Transforms

Crossrefs

Invert A002896, A094059.

Column k=3 of A361397.

Programs

Maple

read transforms; t1 := [ seq(A002896(i),i=1..25) ]; INVERTi(t1);
# second Maple program:
b:= proc(n) option remember; `if`(n<2, 5*n+1,
      (2*(2*n-1)*(10*n^2-10*n+3) *b(n-1)
       -36*(n-1)*(2*n-1)*(2*n-3) *b(n-2)) /n^3)
    end:
g:= proc(n) g(n):= `if` (n<1, -1, -add(g(n-i) *b(i), i=1..n)) end:
a:= n-> abs(g(n)):
seq(a(n), n=0..30);  # Alois P. Heinz, Nov 02 2012

Mathematica

(* A002896 : *) b[n_] := b[n] = Binomial[2*n, n]*HypergeometricPFQ[{1/2, -n, -n}, {1, 1}, 4]; max = 32; a[0] = 1; se = Series[ Sum[ a[n] x^(2 n), {n, 1, max}] - 1 + 1/Sum[ b[n]*x^(2 n), {n, 0, max}], {x, 0, max}]; coes = CoefficientList[se, x]; sol = First[ Solve[ Thread[ coes == 0]]]; Table[ a[n], {n, 0, 16}] /. sol (* Jean-François Alcover, Dec 20 2011 *)
b[n_] := b[n] = If[n < 2, 5*n + 1, (2*(2*n - 1)*(10*n^2 - 10*n + 3)*b[n-1] - 36*(n - 1)*(2*n - 1)*(2*n - 3)*b[n-2]) / n^3];
g[n_] := g[n] = If[n < 1, -1, -Sum [g[n - i]*b[i], {i, 1, n}]];
a[n_] := Abs[g[n]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jan 12 2018, after Alois P. Heinz *)

Formula

Define a_0, a_1, ... = [ 1, 6, 54, ... ] by 1+Sum b_i x^i = 1/(1-Sum a_i x^i) where b_0, b_1, ... = [ 1, 6, 90, ... ] = A002896.

Or, Sum[ a(n) x^(2n), n=1, 2, ...infinity ] = 1-1/Sum[ A002896(n)*x^(2n), n=0, 1, ...infinity ].

G.f.: 2-sqrt(1+12*z) /hypergeom([1/8, 3/8], [1], 64/81*z *(1+sqrt(1-36*z))^2 *(2+sqrt(1-36*z))^4 /(1+12*z)^4)/ hypergeom([1/8, 3/8], [1], 64/81*z *(1-sqrt(1-36*z))^2 *(2-sqrt(1-36*z))^4 /(1+12*z)^4). - Sergey Perepechko, Jan 30 2011

a(n) ~ c * 36^n / n^(3/2), where c = 0.1014559485279103938501072426734... . - Vaclav Kotesovec, Sep 13 2014

c = 384 * (3 + 2*sqrt(3)) * Pi^(9/2) / (Gamma(1/24)^4 * Gamma(11/24)^4). - Vaclav Kotesovec, Apr 23 2023