|
%I #83 Jan 10 2022 11:43:33
%S 1,5,27,155,931,5775,36645,236325,1542195,10153775,67313377,448691985,
%T 3004182349,20188647185,136094684907,919884469275,6232016686995,
%U 42305974804575,287706424085745,1959685788407025,13367193276457881,91295551930615005,624255065007468207
%N Expansion of 1/(sqrt(1-3*x)*sqrt(1-7*x)).
%C Binomial transform of A081671. 3rd binomial transform of A000984. Binomial transform is A098410.
%C Largest coefficient of (1+5*x+x^2)^n; row sums of triangle in A126331. - _Philippe Deléham_, Oct 02 2007
%C 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 come in five colors. - _N-E. Fahssi_, Feb 05 2008
%C 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 five colors. - _N-E. Fahssi_, Mar 31 2008
%C Diagonal of rational function 1/(1 - (x^2 + 5*x*y + y^2)). - _Gheorghe Coserea_, Aug 01 2018
%H Seiichi Manyama, <a href="/A098409/b098409.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H 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 Isaac DeJager, Madeleine Naquin and Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H 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], 2011-2012 (the sequence b_{5,n}).
%H 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 G.f.: 1/sqrt(1-10*x+21*x^2).
%F E.g.f.: exp(5x)*BesselI(0, 2x).
%F a(n) = Sum_{k=0..n} 3^(n-k)*binomial(n,k)*binomial(2k,k). - _Paul Barry_, Mar 08 2005
%F a(n) = [x^n] (1+5*x+x^2)^n. - _Emanuele Munarini_, Apr 27 2012
%F D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 21*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 15 2012
%F a(n) ~ 7^(n+1/2)/(2*sqrt(Pi*n)). - _Vaclav Kotesovec_, Oct 15 2012
%F a(n) = Sum_{k=0..n} 7^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). - _Seiichi Manyama_, Apr 22 2019
%F a(n) = Sum_{k=0..floor(n/2)} 5^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - _Seiichi Manyama_, May 04 2019
%F From _Peter Bala_, Jan 10 2022: (Start)
%F exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + 5*x + 26*x^2 + 140*x^3 + 777*x^4 + ... is the o.g.f. of A182401.
%F 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 a(n) = (1/Pi) * Integral_{x = -1..1} (3 + 4*x^2)^n/sqrt(1 - x^2) dx = (1/Pi) * Integral_{x = -1..1} (7 - 4*x^2)^n/sqrt(1 - x^2) dx. (End)
%t Table[SeriesCoefficient[1/(Sqrt[1-3*x]*Sqrt[1-7*x]),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 15 2012 *)
%t CoefficientList[Series[1/(Sqrt[1-3x] Sqrt[1-7x]),{x,0,30}],x] (* _Harvey P. Dale_, Jun 20 2015 *)
%o (Maxima) a(n):=coeff(expand((1+5*x+x^2)^n),x^n);
%o makelist(a(n),n,0,30); /* _Emanuele Munarini_, Apr 27 2012 */
%o (PARI) x='x+O('x^66); Vec(1/(sqrt(1-3*x)*sqrt(1-7*x))) \\ _Joerg Arndt_, May 11 2013
%o (PARI) {a(n) = sum(k=0, n, 7^(n-k)*(-1)^k*binomial(n, k)*binomial(2*k, k))} \\ _Seiichi Manyama_, Apr 22 2019
%o (PARI) {a(n) = sum(k=0, n\2, 5^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k))} \\ _Seiichi Manyama_, May 04 2019
%Y Column 5 of A292627. Cf. A182401.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 07 2004
|
