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%I #104 May 31 2023 09:11:29
%S 1,5,33,245,1921,15525,127905,1067925,9004545,76499525,653808673,
%T 5614995765,48416454529,418895174885,3634723102113,31616937184725,
%U 275621102802945,2407331941640325,21061836725455905,184550106298084725
%N Coefficients of expansion of 1/sqrt(1 - 10*x + 9*x^2); also, a(n) is the central coefficient of (1 + 5*x + 4*x^2)^n.
%C Also number of paths from (0,0) to (n,0) using steps U=(1,1), H=(1,0) and D=(1,-1), the U steps come in four colors and the H steps come in five colors. - _N-E. Fahssi_, Mar 30 2008
%C Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and three kinds of steps (1,1). - _Joerg Arndt_, Jul 01 2011
%C Sums of squares of coefficients of (1+2*x)^n. - _Joerg Arndt_, Jul 06 2011
%C The Hankel transform of this sequence gives A103488. - _Philippe Deléham_, Dec 02 2007
%C Partial sums of A085363. - _J. M. Bergot_, Jun 12 2013
%C Diagonal of rational functions 1/(1 - x - y - 3*x*y), 1/(1 - x - y*z - 3*x*y*z). - _Gheorghe Coserea_, Jul 06 2018
%H Seiichi Manyama, <a href="/A084771/b084771.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Vincenzo Librandi)
%H T. Amdeberhan, <a href="http://mathoverflow.net/questions/271037/">In search of multiple expressions for a sequence</a>
%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8.
%H Hacène Belbachir and Abdelghani Mehdaoui, <a href="https://doi.org/10.2989/16073606.2020.1729269">Recurrence relation associated with the sums of square binomial coefficients</a>, Quaestiones Mathematicae (2021) Vol. 44, Issue 5, 615-624.
%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 Curtis Greene, <a href="https://doi.org/10.1016/0097-3165(88)90018-0">Posets of shuffles</a>, Journal of Combinatorial Theory, Series A 47.2 (1988): 191-206. See Eq. (30).
%H Christopher Huffaker, Nathan McCue, Cameron N. Miller, and Kayla S. Miller, <a href="http://arxiv.org/abs/1508.06542">The M&M Game: From Morsels to Modern Mathematics</a>, arXiv:1508.06542 [math.HO], 2015.
%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.
%H Michael Z. Spivey and Laura L. Steil, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F G.f.: 1 / sqrt(1 - 10*x + 9*x^2).
%F From _Vladeta Jovovic_, Aug 20 2003: (Start)
%F Binomial transform of A059304.
%F G.f.: Sum_{k >= 0} binomial(2*k,k)*(2*x)^k/(1-x)^(k+1).
%F E.g.f.: exp(5*x)*BesselI(0, 4*x). (End)
%F a(n) = Sum_{k = 0..n} Sum_{j = 0..n-k} C(n,j)*C(n-j,k)*C(2*n-2*j,n-j). - _Paul Barry_, May 19 2006
%F a(n) = Sum_{k = 0..n} 4^k*C(n,k)^2. - heruneedollar (heruneedollar(AT)gmail.com), Mar 20 2010
%F a(n) ~ 3^(2*n+1)/(2*sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Sep 11 2012
%F D-finite with recurrence: n*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-1)*a(n-2). - _R. J. Mathar_, Nov 26 2012
%F a(n) = hypergeom([-n, -n], [1], 4). - _Vladimir Reshetnikov_, Nov 29 2013
%F a(n) = hypergeom([-n, 1/2], [1], -8). - _Peter Luschny_, Apr 26 2016
%F From _Michael Somos_, Jun 01 2017: (Start)
%F a(n) = -3 * 9^n * a(-1-n) for all n in Z.
%F 0 = a(n)*(+81*a(n+1) -135*a(n+2) +18*a(n+3)) +a(n+1)*(-45*a(n+1) +100*a(n+2) -15*a(n+3)) +a(n+2)*(-5*a(n+2) +a(n+3)) for all n in Z. (End)
%F From _Peter Bala_, Nov 13 2022: (Start)
%F 1 + x*exp(Sum_{n >= 1} a(n)*x^n/n) = 1 + x + 5*x^2 + 29*x^3 + 185*x^4 + 1257*x^5 + ... is the g.f. of A059231.
%F The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all positive integers n and r and all primes p. (End)
%F a(n) = 3^n * LegendreP(n, 5/3). - _G. C. Greubel_, May 30 2023
%e G.f.: 1/sqrt(1-2*b*x+(b^2-4*c)*x^2) yields central coefficients of (1+b*x+c*x^2)^n.
%p seq(simplify(hypergeom([-n,1/2], [1], -8)),n=0..19); # _Peter Luschny_, Apr 26 2016
%t Table[n! SeriesCoefficient[E^(5 x) BesselI[0, 4 x], {x, 0, n}], {n, 0, 30}] (* _Vincenzo Librandi_, May 10 2013 *)
%t Table[Hypergeometric2F1[-n, -n, 1, 4], {n,0,30}] (* _Vladimir Reshetnikov_, Nov 29 2013 *)
%t CoefficientList[Series[1/Sqrt[1-10x+9x^2],{x,0,30}],x] (* _Harvey P. Dale_, Mar 08 2016 *)
%t Table[3^n*LegendreP[n, 5/3], {n, 0, 40}] (* _G. C. Greubel_, May 30 2023 *)
%o (PARI) {a(n) = if( n<0, -3 * 9^n * a(-1-n), sum(k=0,n, binomial(n, k)^2 * 4^k))}; /* _Michael Somos_, Oct 08 2003 */
%o (PARI) {a(n) = if( n<0, -3 * 9^n * a(-1-n), polcoeff((1 + 5*x + 4*x^2)^n, n))}; /* _Michael Somos_, Oct 08 2003 */
%o (PARI) /* as lattice paths: same as in A092566 but use */
%o steps=[[1,0], [0,1], [1,1], [1,1], [1,1]]; /* note the triple [1,1] */
%o /* _Joerg Arndt_, Jul 01 2011 */
%o (PARI) a(n)={local(v=Vec((1+2*x)^n));sum(k=1,#v,v[k]^2);} /* _Joerg Arndt_, Jul 06 2011 */
%o (PARI) a(n)={local(v=Vec((1+2*I*x)^n)); sum(k=1,#v, real(v[k])^2+imag(v[k])^2);} /* _Joerg Arndt_, Jul 06 2011 */
%o (GAP) List([0..20],n->Sum([0..n],k->Binomial(n,k)^2*4^k)); # _Muniru A Asiru_, Jul 29 2018
%o (Magma) [3^n*Evaluate(LegendrePolynomial(n), 5/3) : n in [0..40]]; // _G. C. Greubel_, May 30 2023
%o (SageMath) [3^n*gen_legendre_P(n, 0, 5/3) for n in range(41)] # _G. C. Greubel_, May 30 2023
%Y Cf. A001850, A059231, A059304, A246923 (a(n)^2).
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Jun 10 2003
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