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A240721 Expansion of -(4*x+sqrt(1-8*x)-1)/(sqrt(1-8*x)*(4*x^2+x)+8*x^2-x). 1
1, 7, 49, 351, 2561, 18943, 141569, 1066495, 8085505, 61616127, 471556097, 3621830655, 27902803969, 215530668031, 1668644405249, 12944666918911, 100598145875969, 783027553697791, 6103529011806209, 47636654222999551, 372225072921837569, 2911581699143892991 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms 0..200 from Vincenzo Librandi)

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( 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) 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

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 *)

PROG

(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

CROSSREFS

Sequence in context: A024582 A024587 A351057 * A233658 A344270 A144820

Adjacent sequences:  A240718 A240719 A240720 * A240722 A240723 A240724

KEYWORD

nonn

AUTHOR

Vladimir Kruchinin, Apr 11 2014

STATUS

approved

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Last modified May 23 11:45 EDT 2022. Contains 353975 sequences. (Running on oeis4.)