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A139761
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a(n) = Sum_{k >= 0} binomial(n,5*k+4).
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18
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0, 0, 0, 0, 1, 5, 15, 35, 70, 127, 220, 385, 715, 1430, 3004, 6385, 13380, 27370, 54740, 107883, 211585, 416405, 826045, 1652090, 3321891, 6690150, 13455325, 26985675, 53971350, 107746282, 214978335, 429124630, 857417220, 1714834440, 3431847189
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OFFSET
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0,6
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COMMENTS
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Sequence is identical to its fifth differences. - Paul Curtz, Jun 18 2008
{A139398, A133476, A139714, A139748, A139761} is the difference analog of the hyperbolic functions of order 5, {h_1(x), h_2(x), h_3(x), h_4(x), h_5 (x)}. For a definition see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jun 28 2017
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REFERENCES
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A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
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LINKS
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FORMULA
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a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5).
G.f.: x^4/((1-2*x)*(1-3*x+4*x^2-2*x^3+x^4)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
a(n) = round((2/5)*(2^(n-1) + phi^n*cos(Pi*(n-8)/5))), where phi is the golden ratio, round(x) is the integer nearest to x. - Vladimir Shevelev, Jun 28 2017
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MAPLE
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a:= n-> (Matrix(5, (i, j)-> `if`((j-i) mod 5 in [0, 1], 1, 0))^n)[2, 1]:
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MATHEMATICA
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CoefficientList[Series[x^4/((1-2x)(x^4-2x^3+4x^2-3x+1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 21 2015 *)
LinearRecurrence[{5, -10, 10, -5, 2}, {0, 0, 0, 0, 1}, 35] (* Jean-François Alcover, Feb 14 2018 *)
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PROG
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(PARI) a(n) = sum(k=0, n\5, binomial(n, 5*k+4)); \\ Michel Marcus, Dec 21 2015
(PARI) my(x='x+O('x^100)); concat([0, 0, 0, 0], Vec(-x^4/((2*x-1)*(x^4-2*x^3 +4*x^2-3*x+1)))) \\ Altug Alkan, Dec 21 2015
(Magma) I:=[0, 0, 0, 0, 1]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+2*Self(n-5): n in [1..40]]; // Vincenzo Librandi, Dec 21 2015
(SageMath)
def A139761(n): return sum(binomial(n, 5*k+4) for k in range(1+n//5))
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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