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A098410 Expansion of 1/(sqrt(1-4*x)*sqrt(1-8*x)).

%I A098410 #92 May 20 2025 10:22:25
%S A098410 1,6,38,252,1734,12276,88796,652728,4856902,36478404,275975028,
%T A098410 2099978568,16054486044,123213933576,948713646072,7325088811632,
%U A098410 56692748053062,439689331938276,3416328042565124,26587566855421608,207218159714453044,1617124976299315224,12634892752595949192
%N A098410 Expansion of 1/(sqrt(1-4*x)*sqrt(1-8*x)).
%C A098410 Convolution of A000984(n) and 2^n*A000984(n). Convolution of A000984(n) and A059304. 4th binomial transform of A000984.
%C A098410 Largest coefficient of (1 + 6*x + x^2)^n. - _Philippe Deléham_, Oct 02 2007
%C A098410 Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the H steps can have 6 colors. - _N-E. Fahssi_, Mar 31 2008
%C A098410 Self-convolution of a(n)/4^n gives A126646. - _Vladimir Reshetnikov_, Oct 10 2016
%C A098410 Diagonal of rational function 1/(1 - (x^2 + 6*x*y + y^2)). - _Gheorghe Coserea_, Aug 03 2018
%H A098410 Seiichi Manyama, <a href="/A098410/b098410.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H A098410 Hacène Belbachir, Abdelghani Mehdaoui and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Szalay/szalay42.html">Diagonal Sums in the Pascal Pyramid, II: Applications</a>, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.
%H A098410 Francesc Fite, Kiran S. Kedlaya, Victor Rotger and Andrew V. Sutherland, <a href="http://arxiv.org/abs/1110.6638">Sato-Tate distributions and Galois endomorphism modules in genus 2</a>, arXiv preprint arXiv:1110.6638 [math.NT], 2011 (the sequence b_{6,n}).
%H A098410 Rigoberto Flórez, Leandro Junes and José L. Ramírez, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Florez/florez4.html">Further Results on Paths in an n-Dimensional Cubic Lattice</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.2.
%H A098410 Tony D. Noe, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Noe/noe35.html">On the Divisibility of Generalized Central Trinomial Coefficients</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.
%F A098410 G.f.: 1/sqrt(1 - 12*x + 32*x^2).
%F A098410 E.g.f.: exp(6*x)*BesselI(0, 2*x).
%F A098410 a(n) = Sum_{k=0..n} 2^k*binomial(2*k, k)*binomial(2*(n-k), n-k).
%F A098410 a(n) = Sum_{k=0..n} 4^(n-k)*binomial(n,k)*binomial(2k,k). - _Paul Barry_, Mar 08 2005
%F A098410 D-finite with recurrence: n*a(n) = 6*(2*n-1)*a(n-1) - 32*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012
%F A098410 a(n) ~ 2^(3*n+1/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 15 2012
%F A098410 a(n) = 4^n*hypergeometric([-n, 1/2], [1], -1). - _Peter Luschny_, May 19 2015
%F A098410 a(n) = Sum_{k=0..n} 8^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - _Seiichi Manyama_, Apr 22 2019
%F A098410 a(n) = Sum_{k=0..floor(n/2)} 6^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - _Seiichi Manyama_, May 04 2019
%F A098410 From _Peter Bala_, Jan 10 2022: (Start)
%F A098410 3*x + x^2*exp(Sum_{n >= 1} a(n)*x^n/n) = 3*x + x^2 + 6*x^3 + 37*x^4 + 234*x^5 + 1514*x^6 + ... is the o.g.f. of A025230.
%F A098410 The Gauss congruences a(n*p^k) == a(n*p^(k-1)) (mod p^k) hold for prime p and positive integers n and k.
%F A098410 a(n) = (1/Pi) * Integral_{x = -1..1} (4 + 4*x^2)^n/sqrt(1 - x^2) dx  = (1/Pi) * Integral_{x = -1..1} (8 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)
%e A098410 G.f. = 1 + 6*x + 38*x^2 + 252*x^3 + 1734*x^4 + 12276*x^5 + 88796*x^6 + ...
%t A098410 Table[SeriesCoefficient[1/(Sqrt[1-4*x]*Sqrt[1-8*x]),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2012 *)
%t A098410 a[ n_] := If[n < 0, 0, 4^n Hypergeometric2F1[-n, 1/2, 1, -1]]; (* _Michael Somos_, May 06 2017 *)
%t A098410 a[ n_] := SeriesCoefficient[ D[ InverseJacobiSD[2 x, -1] / 2, x], {x, 0, 2 n}]; (* _Michael Somos_, May 06 2017 *)
%o A098410 (PARI) x='x+O('x^66); Vec(1/sqrt(1-12*x+32*x^2)) \\ _Joerg Arndt_, May 11 2013
%o A098410 (PARI) {a(n) = sum(k=0, n, 8^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ _Seiichi Manyama_, Apr 22 2019
%o A098410 (PARI) {a(n) = sum(k=0, n\2, 6^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ _Seiichi Manyama_, May 04 2019
%o A098410 (Sage)
%o A098410 a = lambda n: 4^n*hypergeometric([-n, 1/2], [1], -1)
%o A098410 [simplify(a(n)) for n in range(23)] # _Peter Luschny_, May 19 2015
%Y A098410 Column 6 of A292627. Cf. A025230, A104454 (binomial transf.)
%K A098410 easy,nonn
%O A098410 0,2
%A A098410 _Paul Barry_, Sep 07 2004