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A049037
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Number of cubic lattice walks that start and end at origin after 2n steps, not touching origin at intermediate stages.
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9
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1, 6, 54, 996, 22734, 577692, 15680628, 445162392, 13055851998, 392475442092, 12029082873372, 374482032292008, 11808861461931492, 376406128925067528, 12108063535794336312, 392560994063887113744, 12814685828476778001726, 420836267423433182275404
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OFFSET
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0,2
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REFERENCES
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S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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)):
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MATHEMATICA
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(* 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]];
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CROSSREFS
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KEYWORD
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easy,nonn,nice
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AUTHOR
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Alessandro Zinani (alzinani(AT)tin.it)
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STATUS
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approved
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