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%I #49 Jan 21 2023 01:28:29
%S 1,7,49,351,2561,18943,141569,1066495,8085505,61616127,471556097,
%T 3621830655,27902803969,215530668031,1668644405249,12944666918911,
%U 100598145875969,783027553697791,6103529011806209,47636654222999551,372225072921837569,2911581699143892991
%N Expansion of -(4*x + sqrt(1-8*x) - 1)/(sqrt(1-8*x)*(4*x^2+x) + 8*x^2 - x).
%H G. C. Greubel, <a href="/A240721/b240721.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, <a href="https://arxiv.org/abs/2301.02628">Pinnacle sets of signed permutations</a>, arXiv:2301.02628 [math.CO], 2023.
%F a(n) = Sum_{k=0..n} (k+1)*binomial(2*(n+1),n-k)*binomial(n+k+1,n))/(n+1).
%F a(n) = Sum_{k=0..n} binomial(2*(n+1),k)*2^k*(-1)^(n+k) = binomial(2*(n+1),n+1)*(n+1)*Sum_{k=0..n} binomial(n,k)/(n+k+2). - _Max Alekseyev_, Jun 16 2021
%F A(x) = (x*B'(x)+B(x))/(x*B(x)+1) where B(x) = (1-4*x-sqrt(1-8*x))/(8*x^2) is the g.f. of A003645.
%F a(n) ~ 2^(3*n+3)/(3*sqrt(Pi*n)). - _Vaclav Kotesovec_, Apr 12 2014
%F a(n) = C(2*n+2, n)*2F1([-n, n+2], [n+3], -1), 2F1 is the hypergeometric function. - _Peter Luschny_, Jul 16 2014
%F a(n) = 8*Sum_{k=0..n-3} a(k)*a(n-3-k) + 7*Sum_{k=0..n-2} a(k)*a(n-2-k) - Sum_{k=0..n-1} a(k)*a(n-1-k) + 8*a(n-1) for n > 2, a(0)=1, a(1)=7, a(2)=49. - _Tani Akinari_, Jul 16 2014
%F D-finite with recurrence -(n+1)*(3*n-2)*a(n) +(21*n^2-5*n-2)*a(n-1) +4*(3*n+1)*(2*n-1)*a(n-2)=0. - _R. J. Mathar_, Jun 14 2016
%p a := n -> binomial(2*n+2,n)*hypergeom([-n, n+2], [n+3],-1);
%p seq(round(evalf(a(n), 32)), n=0..19); # _Peter Luschny_, Jul 16 2014
%t CoefficientList[Series[-(4 x + Sqrt[1 - 8 x] - 1)/(Sqrt[1 - 8 x] (4 x^2 + x) + 8 x^2 - x), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 12 2014 *)
%o (Maxima)
%o a(n) := sum((k+1)*binomial(2*(n+1), n-k)*binomial(n+k+1,n), k, 0, n)/(n+1);
%o (Maxima) a[0]:1$ a[1]:7$ a[2]:49$ a[n] := 8*sum(a[k]*a[n-3-k], k, 0, n-3)+7*sum(a[k]*a[n-2-k], k, 0, n-2)-sum(a[k]*a[n-1-k], k, 0, n-1)+8*a[n-1]$ makelist(a[n], n, 0, 1000); /* _Tani Akinari_, Jul 16 2014 */
%o (PARI) x='x+O('x^50); Vec(-(4*x+sqrt(1-8*x)-1)/(sqrt(1-8*x)*(4*x^2+x)+8*x^2-x)) \\ _G. C. Greubel_, Apr 05 2017
%K nonn
%O 0,2
%A _Vladimir Kruchinin_, Apr 11 2014
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