A139761 a(n) = Sum_{k >= 0} binomial(n,5*k+4).

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

Offset: 0

Author

N. J. A. Sloane, Jun 13 2008

Keywords

• nonn
• easy

Comments

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

This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^5; see A291000. - Clark Kimberling, Aug 24 2017

References

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• 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).

Crossrefs

Cf. A049016, A133476, A139398, A139714, A139748, A139761.

Programs

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

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/((1-2x)(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 *)

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

SageMath

def A139761(n): return sum(binomial(n,5*k+4) for k in range(1+n//5))
[A139761(n) for n in range(41)] # G. C. Greubel, Jan 23 2023

Formula

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

From Paul Curtz, Jun 18 2008: (Start)

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5).

a(n) = A139398(n+1) - A139398(n). (End)

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

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