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%I #59 Jan 02 2023 12:30:46
%S 0,0,1,3,6,10,14,14,0,-48,-164,-396,-792,-1352,-1912,-1912,0,6528,
%T 22288,53808,107616,183712,259808,259808,0,-887040,-3028544,-7311552,
%U -14623104,-24963200,-35303296,-35303296,0,120532992,411525376,993510144,1987020288,3392055808,4797091328,4797091328
%N Coefficient of x^2 in (1+x)^n mod 1+x^4.
%C {A099586, A099587, A099588, A099589) is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x)} of order 4. For the definition, see [Erdelyi] and the Shevelev link. - _Vladimir Shevelev_, Jul 04 2017
%D A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.
%H G. C. Greubel, <a href="/A099588/b099588.txt">Table of n, a(n) for n = 0..1000</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 Vladimir Shevelev, <a href="http://list.seqfan.eu/oldermail/seqfan/2017-July/017798.html">Coefficient of x^k in ((x+1)^n modulo x^N+1)</a>, seqfan, Thu Jul 20 2017.
%H G. Tollisen and T. Lengyel, <a href="http://www.emis.de/journals/INTEGERS/papers/e4/e4.Abstract.html">A congruential identity and the 2-adic order of lacunary sums of binomial coefficients</a>, Integers 4 (2004), #A4.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-2).
%F G.f.: -x^2*(x-1) / (2*x^4-4*x^3+6*x^2-4*x+1). - _Colin Barker_, Jul 15 2013
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-4). - _G. C. Greubel_, Nov 10 2015
%F a(n) = (1/2)*((2+sqrt(2))^(n/2) * sin(n*Pi/8) - (2-sqrt(2))^(n/2)*sin(3*n*Pi/8)). - _G. C. Greubel_, Nov 10 2015
%F From _Vladimir Shevelev_, Jul 04 2017 (Start)
%F a(n) = Sum_{k>=0}(-1)^k*binomial(n,4*k+2);
%F a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-4)/8)/2), where round(x) is the integer nearest to x;
%F a(n+m) = a(n)*K_1(m) + K_2(n)*K_2(m) + K_1(n)*a(m) - K_4(n)*K_4(m), where K_1 is A099586, K_2 is A099587, K_4 is A099589. (End)
%p f:= rectoproc({rec, a(0)=0,a(1)=0,a(2)=1,a(3)=3},a(n),remember):
%p map(f, [$0..100]); # _Robert Israel_, Jun 30 2017
%t RecurrenceTable[{a[n]==4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4], a[1]=0, a[2]=1, a[3]=3, a[4]=6}, a, {n, 1, 200}] (* _G. C. Greubel_, Nov 10 2015 *)
%t Table[Sum[(-1)^k*Binomial[n, 4 k + 2], {k, 0, n}], {n, 0, 36}] (* _Michael De Vlieger_, Jun 30 2017 *)
%t a[n_] := n*(n-1)/2 HypergeometricPFQ[{(2-n)/4, (3-n)/4, (4-n)/4, (5-n)/4}, {3/4, 5/4, 3/2}, -1]; Array[a, 40, 0] (* _Jean-François Alcover_, Jul 20 2017, from _Vladimir Shevelev_'s first formula *)
%o (PARI) x='x+O('x^55); concat([0, 0], Vec(-x^2*(x-1)/(2*x^4-4*x^3+6*x^2-4*x+1))) \\ _Altug Alkan_, Nov 11 2015
%o (PARI) a(n) = sum(t=0, (n-2)\4, (-1)^t*binomial(n,4*t+2)); \\ _Michel Marcus_, Jun 30 2017
%o (PARI) a(n)=polcoeff(lift(Mod(1+x,1+x^4)^n),2); \\ _Joerg Arndt_, Feb 22 2018
%Y Cf. A099586, A099587, A099589.
%K sign,easy
%O 0,4
%A _Ralf Stephan_, Oct 24 2004
%E a(0)=0 added by _N. J. A. Sloane_, Jul 04 2017
%E a(673) in b-file corrected by _Andrew Howroyd_, Feb 21 2018
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