A181933 a(n) = Sum_{k=0..n} binomial(n+k,k)*sin(Pi*(n+k)/2).
0, 1, -3, 9, -30, 106, -385, 1421, -5304, 19966, -75658, 288222, -1102790, 4234868, -16312773, 63003869, -243896960, 946066678, -3676303578, 14308370014, -55768166380, 217640082188, -850345208538, 3325907590274, -13020993588680
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Mathematica
f[n_] := Sum[ Binomial[n + k, k] Sin[Pi (n + k)/2], {k, 0, n}]; Array[f, 25, 0] -
Maxima
makelist(coeff(taylor(1/2*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1),x,0,20),x,n),n,0,20); /* Vladimir Kruchinin, Mar 28 2016 */
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PARI
x='x+O('x^50); concat([0], Vec((1/2)*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1))) \\ G. C. Greubel, Mar 24 2017
Formula
G.f.: (1/2)*(sqrt(4*x+1)*(1+x)-3*x-1)/(sqrt(4*x+1)*(x^2+3*x+1)-4*x^2-5*x-1). - Vladimir Kruchinin, Mar 28 2016
a(n) ~ (-1)^(n+1) *2^(2*n+1) / (5*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 28 2016
Conjecture: +2*n*a(n) +8*n*a(n-1) +(-n+20)*a(n-2) +5*(-n+4)*a(n-3) +2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Jun 14 2016