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A049037 Number of cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages. 5
1, 6, 54, 996, 22734, 577692, 15680628, 445162392, 13055851998, 392475442092, 12029082873372, 374482032292008, 11808861461931492, 376406128925067528, 12108063535794336312, 392560994063887113744, 12814685828476778001726 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice

N. J. A. Sloane, Transforms

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

EXAMPLE

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.

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-Fran├žois Alcover, Dec 20 2011 *)

CROSSREFS

Invert A002896.

Sequence in context: A217238 A171681 A267837 * A047681 A075575 A073655

Adjacent sequences:  A049034 A049035 A049036 * A049038 A049039 A049040

KEYWORD

easy,nonn,nice

AUTHOR

Alessandro Zinani (alzinani(AT)tin.it)

STATUS

approved

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Last modified June 22 14:36 EDT 2017. Contains 288633 sequences.