A139748 a(n) = Sum_{ k >= 0} binomial(n,5*k+3).

0, 0, 0, 1, 4, 10, 20, 35, 57, 93, 165, 330, 715, 1574, 3381, 6995, 13990, 27370, 53143, 103702, 204820, 409640, 826045, 1669801, 3368259, 6765175, 13530350, 26985675, 53774932, 107232053, 214146295, 428292590, 857417220, 1717012749, 3437550076

Offset: 0

Author

N. J. A. Sloane, Jun 13 2008

Keywords

• nonn
• easy

Comments

From Gary W. Adamson, Mar 14 2009: (Start)

M^n * [1,0,0,0,0] = [A139398(n), A139761(n), a(n), A139714(n), A133476(n)]

where M = a 5 X 5 matrix [1,1,0,0,0; 0,1,1,0,0; 0,0,1,1,0; 0,0,0,1,1; 1,0,0,0,1].

Sum of terms = 2^n. Example: M^6 * [1,0,0,0,0] = [7, 15, 20, 15, 7]; sum = 64. (End)

{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

• 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, A139714.

Programs

Magma

I:=[0,0,0,1,4]; [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)[3, 1]:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 21 2015

Mathematica

CoefficientList[Series[x^3 (x - 1)/((2 x - 1) (x^4 - 2 x^3 + 4 x^2 - 3 x + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

PARI

a(n) = sum(k=0, n\5, binomial(n,5*k+3)); \\ Michel Marcus, Dec 21 2015

PARI

x='x+O('x^100); concat([0, 0, 0], Vec(x^3*(x-1)/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)))) \\ Altug Alkan, Dec 21 2015

Formula

G.f.: x^3*(x-1)/((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) = round((2/5)*(2^(n-1) + phi^n*cos(Pi*(n-6)/5))), where phi is the golden ratio and round(x) is the integer nearest to x. - Vladimir Shevelev, Jun 28 2017

a(n+m) = a(n)*H_1(m) + H_3(n)*H_2(m) + H_2(n)*H_3(m) + H_1(n)*a(m) + H_5(n)*H_5(m), where H_1=A139398, H_2=A133476, H_3=A139714, H_5=A139761. - Vladimir Shevelev, Jun 28 2017