|
%I #50 Oct 21 2022 22:03:16
%S 0,0,1,3,6,10,15,22,36,72,165,385,859,1807,3614,6995,13380,25773,
%T 50559,101118,204820,416405,843756,1698458,3396916,6765175,13455325,
%U 26789257,53457121,106914242,214146295,429124630,859595529,1720537327,3441074654
%N a(n) = Sum_{k>=0} binomial(n,5*k+2).
%C From _Gary W. Adamson_, Mar 14 2009: (Start)
%C M^n * [1,0,0,0,0] = [A139398(n), A139761(n), A139748(n), a(n), A133476(n)]
%C 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].
%C Sum of terms = 2^n. Example: M^6 = [7, 15, 20, 15, 7], sum = 2^6 = 64. (End)
%C {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 18 2017
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H Seiichi Manyama, <a href="/A139714/b139714.txt">Table of n, a(n) for n = 0..3000</a>
%H Vladimir Shevelev, <a href="https://arxiv.org/abs/1706.01454">Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n</a>, arXiv:1706.01454 [math.CO], 2017.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,2).
%F G.f.: -x^2*(x-1)^2/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)). - Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009
%F a(n) = round((2/5)*(2^(n-1)+phi^n*cos(Pi*(n-4)/5))), where phi is the golden ratio, round(x) is the integer nearest to x. - _Vladimir Shevelev_, Jun 18 2017
%F a(n+m) = a(n)*H_1(m) + H_2(n)*H_2(m) + H_1(n)*a(m) + H_5(n)*H_4(m) + H_4(n)*H_5(m), where H_1=A139398, H_2=A133476, H_4=A139748, H_5=A139761. - _Vladimir Shevelev_, Jun 18 2017
%p a:= n-> (Matrix(5, (i, j)-> `if`((j-i) mod 5 in [0, 1], 1, 0))^n)[4, 1]:
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Dec 21 2015
%t CoefficientList[Series[x^2 (x - 1)^2/((1 - 2 x) (x^4 - 2 x^3 + 4 x^2 - 3 x + 1)), {x, 0, 50}], x] (* _Vincenzo Librandi_, Dec 21 2015 *)
%o (PARI) a(n) = sum(k=0, n\5, binomial(n,5*k+2)); \\ _Michel Marcus_, Dec 21 2015
%o (PARI) x='x+O('x^100); concat([0, 0], Vec(-x^2*(x-1)^2/((2*x-1)*(x^4-2*x^3+4*x^2-3*x+1)))) \\ _Altug Alkan_, Dec 21 2015
%o (Magma) [n le 5 select (n-2)*(n-1)/2 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
%Y Cf. A049016, A133476, A139748, A139761.
%K nonn,easy
%O 0,4
%A _N. J. A. Sloane_, Jun 13 2008
|
