A098411 Expansion of 1/(sqrt(1-4x)sqrt(1-12x)).

1, 8, 72, 704, 7264, 77568, 847104, 9394176, 105334272, 1190899712, 13551235072, 154997784576, 1780378353664, 20522842062848, 237284128063488, 2750571189633024, 31956067676454912, 371997834879172608, 4337957919010062336, 50664706036388069376, 592558533060795039744

Offset: 0

Author

Paul Barry, Sep 07 2004

Keywords

• easy
• nonn

Comments

Nguyen and Taggart (see link) conjecture: det[a(i+j) for i,j=0..n] = b(n)*b(n+1)/2 with b(n) = A139685(n). - Peter Luschny, May 19 2015

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Hacène Belbachir, Abdelghani Mehdaoui, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
• H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; Mentions this sequence. - From _N. J. A. Sloane_, Mar 16 2014

Crossrefs

Cf. A098410, A139685.

Programs

Mathematica

Table[SeriesCoefficient[1/(Sqrt[1-4*x]*Sqrt[1-12*x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 15 2012 *)

PARI

x='x+O('x^66); Vec(1/sqrt(1-16*x+48*x^2)) \\ Joerg Arndt, May 11 2013

Sage

a = lambda n: 4^n*hypergeometric([-n, 1/2], [1], -2)
[simplify(a(n)) for n in range(23)] # Peter Luschny, May 19 2015

Formula

G.f.: 1/sqrt(1-16x+48x^2).

E.g.f.: exp(8x)*BesselI(0, 4x).

a(n) = Sum_{k=0..n} 3^k*binomial(2k, k)*binomial(2(n-k), n-k).

D-finite with recurrence: n*a(n) +8*(1-2*n)*a(n-1) +48*(n-1)*a(n-2)=0. - R. J. Mathar, Sep 26 2012

a(n) ~ sqrt(3)*12^n/sqrt(2*Pi*n). - Vaclav Kotesovec, Oct 15 2012

a(n) = 4^n*hypergeometric([-n, 1/2], [1], -2). - Peter Luschny, May 19 2015