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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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4
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1, 6, 54, 996, 22734, 577692, 15680628, 445162392, 13055851998, 392475442092, 12029082873372, 374482032292008, 11808861461931492, 376406128925067528, 12108063535794336312, 392560994063887113744, 12814685828476778001726
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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REFERENCES
| S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.
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LINKS
| S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice
N. J. A. Sloane, Transforms
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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
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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.
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MAPLE
| read transforms; t1 := [ seq(A002896(i), i=1..25) ]; INVERTi(t1);
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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 (* From Jean-François Alcover, Dec 20 2011 *)
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CROSSREFS
| Invert A002896.
Sequence in context: A167571 A138434 A171681 * A047681 A075575 A073655
Adjacent sequences: A049034 A049035 A049036 * A049038 A049039 A049040
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KEYWORD
| easy,nonn,nice
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AUTHOR
| Alessandro Zinani (alzinani(AT)tin.it)
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