%I #50 Jan 28 2023 12:34:21
%S 0,0,0,0,1,5,15,35,70,127,220,385,715,1430,3004,6385,13380,27370,
%T 54740,107883,211585,416405,826045,1652090,3321891,6690150,13455325,
%U 26985675,53971350,107746282,214978335,429124630,857417220,1714834440,3431847189
%N a(n) = Sum_{k >= 0} binomial(n,5*k+4).
%C Sequence is identical to its fifth differences. - _Paul Curtz_, Jun 18 2008
%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 28 2017
%C This is the p-INVERT of (1,1,1,1,1,...) for p(S) = 1 - S^5; see A291000. - _Clark Kimberling_, Aug 24 2017
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H Seiichi Manyama, <a href="/A139761/b139761.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 a(n) = A049016(n-4). - _R. J. Mathar_, Nov 08 2010
%F From _Paul Curtz_, Jun 18 2008: (Start)
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5).
%F a(n) = A139398(n+1) - A139398(n). (End)
%F 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
%F 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
%F 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
%p a:= n-> (Matrix(5, (i, j)-> `if`((j-i) mod 5 in [0, 1], 1, 0))^n)[2, 1]:
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Dec 21 2015
%t CoefficientList[Series[x^4/((1-2x)(x^4-2x^3+4x^2-3x+1)), {x,0,40}], x] (* _Vincenzo Librandi_, Dec 21 2015 *)
%t LinearRecurrence[{5,-10,10,-5,2}, {0,0,0,0,1}, 35] (* _Jean-François Alcover_, Feb 14 2018 *)
%o (PARI) a(n) = sum(k=0, n\5, binomial(n,5*k+4)); \\ _Michel Marcus_, Dec 21 2015
%o (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
%o (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
%o (SageMath)
%o def A139761(n): return sum(binomial(n,5*k+4) for k in range(1+n//5))
%o [A139761(n) for n in range(41)] # _G. C. Greubel_, Jan 23 2023
%Y Cf. A049016, A133476, A139398, A139714, A139748, A139761.
%K nonn,easy
%O 0,6
%A _N. J. A. Sloane_, Jun 13 2008