A240721 Expansion of -(4*x + sqrt(1-8*x) - 1)/(sqrt(1-8*x)*(4*x^2+x) + 8*x^2 - x).

1, 7, 49, 351, 2561, 18943, 141569, 1066495, 8085505, 61616127, 471556097, 3621830655, 27902803969, 215530668031, 1668644405249, 12944666918911, 100598145875969, 783027553697791, 6103529011806209, 47636654222999551, 372225072921837569, 2911581699143892991

Offset: 0

Author

Vladimir Kruchinin, Apr 11 2014

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)
• Nicolle González, Pamela E. Harris, Gordon Rojas Kirby, Mariana Smit Vega Garcia, and Bridget Eileen Tenner, Pinnacle sets of signed permutations, arXiv:2301.02628 [math.CO], 2023.

Crossrefs

Cf. A178792.

Programs

Maple

a := n -> binomial(2*n+2,n)*hypergeom([-n, n+2], [n+3],-1);
seq(round(evalf(a(n), 32)), n=0..19); # Peter Luschny, Jul 16 2014

Mathematica

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 *)

Maxima

a(n) := sum((k+1)*binomial(2*(n+1), n-k)*binomial(n+k+1,n), k, 0, n)/(n+1);

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 */

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

Formula

a(n) = (Sum_{k=0..n} (k+1)*binomial(2*(n+1),n-k)*binomial(n+k+1,n))/(n+1).

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

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.

a(n) ~ 2^(3*n+3)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 12 2014

a(n) = C(2*n+2, n)*2F1([-n, n+2], [n+3], -1), 2F1 is the hypergeometric function. - Peter Luschny, Jul 16 2014

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

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

a(n) = 2^(n+1)*binomial(2*n+1,n) - A178792(n). - Akiva Weinberger, Dec 04 2024