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A139761 a(n) = Sum_{ k >= 0} binomial(n,5*k+4). 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

{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

REFERENCES

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

LINKS

Table of n, a(n) for n=0..34.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,2).

FORMULA

a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+2a(n-5). Sequence is identical to its fifth differences. Differences of A139398. Same recurrence holds for differences. Binomial transform of period 5: repeat 0, 0, 0, 0, 1 = A079998(n+1). - Paul Curtz, Jun 18 2008

G.f.: -x^4/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009

a(n) = A049016(n-4). - R. J. Mathar, Nov 08 2010

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

a(n+m) = a(n)*H_1(m) + H_4(n)*H_2(m) + H_3(n)*H_3(m) + H_2(n)*H_4(m) + H_1(n)*a(m), where H_1=A139398, H_2=A133476, H_3=A139714, H_4=A139748. - Vladimir Shevelev, Jun 28 2017

MAPLE

a:= n-> (Matrix(5, (i, j)-> `if`((j-i) mod 5 in [0, 1], 1, 0))^n)[2, 1]:

seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2015

MATHEMATICA

CoefficientList[Series[x^4/((1 - 2 x) (x^4 - 2 x^3 + 4 x^2 - 3 x + 1)), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 21 2015 *)

PROG

(PARI) a(n) = sum(k=0, n\5, binomial(n, 5*k+4)); \\ Michel Marcus, Dec 21 2015

(PARI) 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

CROSSREFS

Cf. A049016, A133476, A139714, A139748.

Sequence in context: A140227 A264925 A049016 * A275935 A137360 A195760

Adjacent sequences:  A139758 A139759 A139760 * A139762 A139763 A139764

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 13 2008

STATUS

approved

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Last modified August 21 00:45 EDT 2017. Contains 290855 sequences.