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A005022
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Number of walks of length 2n+6 in the path graph P_7 from one end to the other.
(Formerly M4171)
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3
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6, 26, 100, 364, 1288, 4488, 15504, 53296, 182688, 625184, 2137408, 7303360, 24946816, 85196928, 290926848, 993379072, 3391793664, 11580678656, 39539651584, 134998297600, 460915984384, 1573671536640, 5372862566400, 18344123969536, 62630804299776
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OFFSET
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1,1
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REFERENCES
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W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed, Wiley, New York, 1968, p. 96.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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G.f.: 1/(1-6x+10x^2-4x^3)-1.
a(k) = sum(binomial(6+2*k, 8*j+k-2)-binomial(6+2*k, 8*j+k-1), j=-infinity..infinity) (a finite sum).
The g.f. x^3/(1-6*x+10*x^2-4*x^3) occurs on page 320 of Viennot, 2002.
a(n) = -2^(1+n)+(3/2-sqrt(2))*(2-sqrt(2))^n+(3/2+sqrt(2))*(2+sqrt(2))^n - Colin Barker, Apr 27 2016
E.g.f.: (-2 + 3*cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x))*exp(2*x) - 1. - Ilya Gutkovskiy, Apr 27 2016
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EXAMPLE
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Example: a(1)=6 because in the path ABCDEFG we have ABABCDEFG, ABCBCDEFG, ABCDCDEFG, ABCDEDEFG, ABCDEFEFG and ABCDEFGFG. - Emeric Deutsch, Apr 02 2004
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MAPLE
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a:=k->sum(binomial(6+2*k, 8*j+k-2), j=ceil((2-k)/8)..floor((8+k)/8))-sum(binomial(6+2*k, 8*j+k-1), j=ceil((1-k)/8)..floor((7+k)/8)): seq(a(k), k=1..28);
A005022:=-1/((2*z-1)*(2*z**2-4*z+1)) -1; # [Conjectured (correctly) by Simon Plouffe in his 1992 dissertation. Gives sequence with an additional leading term of 1.]
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MATHEMATICA
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CoefficientList[Series[(1 / x) (1 / (1 - 6 x + 10 x^2 - 4 x^3) - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 08 2013 *)
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PROG
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(Magma) I:=[6, 26, 100]; [n le 3 select I[n] else 6*Self(n-1)-10*Self(n-2)+4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 08 2013
(PARI) Vec(2*(1-x)*(3-2*x) / ((1-2*x)*(1-4*x+2*x^2)) + O(x^50)) \\ Colin Barker, Apr 27 2016
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CROSSREFS
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KEYWORD
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nonn,walk,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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