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%I #91 Jan 08 2022 22:04:06
%S 1,4,22,136,886,5944,40636,281488,1968934,13875544,98365972,700701808,
%T 5011371964,35961808432,258805997752,1867175631136,13500088649734,
%U 97794850668952,709626281415076,5157024231645616,37528209137458516,273431636191026064,1994448720786816712
%N Define an array as follows: b(i,0)=b(0,j)=1, b(i,j) = 2*b(i-1,j-1) + b(i-1,j) + b(i,j-1). Then a(n) = b(n,n).
%C 2^n*LegendreP(n,k) yields the central coefficients of (1 + 2*k*x + (k^2-1)*x^2)^n, with g.f. 1/sqrt(1 - 4*k*x + 4*x^2) and e.g.f. exp(2*k*x)BesselI(0, 2*sqrt(k^2-1)*x). - _Paul Barry_, May 25 2005
%C Number of Delannoy paths from (0,0) to (n,n) with steps U(0,1), H(1,0) and D(1,1) where D can have two colors. - _Paul Barry_, May 25 2005
%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 can have three colors and H steps can have four colors. - _N-E. Fahssi_, Mar 31 2008
%C Number of lattice paths from (0,0) to (n,n) using steps (1,0), (0,1), and two kinds of steps (1,1). - _Joerg Arndt_, Jul 01 2011
%C Hankel transform is 2^n*3^C(n+1,2) = (-1)^C(n+1,2)*A127946(n). - _Paul Barry_, Jan 24 2011
%C Central terms of triangle A152842. - _Reinhard Zumkeller_, May 01 2014
%C Diagonal of rational functions 1/(1 - x - y - 2*x*y), 1/(1 - x - y*z - 2*x*y*z). - _Gheorghe Coserea_, Jul 06 2018
%C 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. - _Peter Bala_, Jan 07 2022
%D Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.
%H Arkadiusz Wesolowski, <a href="/A069835/b069835.txt">Table of n, a(n) for n = 0..250</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. - _N. J. A. Sloane_, Oct 08 2012
%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 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 Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%F From _Vladeta Jovovic_, May 13 2003: (Start)
%F a(n) = 2^n*LegendreP(n, 2) = 2^n*hypergeom([ -n, n+1], [1], -1/2) = 2^n*GegenbauerC(n, 1/2, 2) = Sum_{k=0..n} 3^k*binomial(n, k)^2.
%F D-finite with recurrence: a(n) = 4*(2*n-1)/n*a(n-1) - 4*(n-1)/n*a(n-2).
%F G.f.: 1/sqrt(1 - 8*x + 4*x^2). (End)
%F a(n) equals the central coefficient of (1 + 4*x + 3*x^2)^n. - _Paul D. Hanna_, Jun 03 2003
%F E.g.f.: exp(4*x)*Bessel_I(0, 2*sqrt(3)*x). - _Paul Barry_, Sep 20 2004
%F a(n) = Sum_{k=0..floor(n/2)} C(n, k)*C(2*(n-k), n)*(-1)^k*2^(n-2*k). - _Paul Barry_, May 25 2005
%F a(n) = Sum_{k=0..n} C(n, k)*C(n+k, k)*2^(n-k). - _Paul Barry_, May 25 2005
%F a(n) = Sum_{k=0..n} C(n, k)^2*3^k. - _Paul Barry_, Oct 15 2005
%F G.f.: 1/(1-4x-6x^2/(1-4x-3x^2/(1-4x-3x^2/(1-4x-3x^2/(1-... (continued fraction). - _Paul Barry_ Jan 24 2011
%F Asymptotic: a(n) ~ sqrt(1/2 + 1/sqrt(3))*(1+sqrt(3))^(2*n)/sqrt(Pi*n). - _Vaclav Kotesovec_, Sep 11 2012
%F 0 = a(n)*(16*a(n+1) - 48*a(n+2) + 8*a(n+3)) + a(n+1)*(-16*a(n+1) + 64*a(n+2) - 12*a(n+3)) + a(n+2)*(-4*a(n+2) + a(n+3)) for all n in Z. - _Michael Somos_, Apr 21 2020
%e The array b is a rewriting of A081577:
%e 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, ...
%e 1, 7, 22, 46, 79, 121, 172, 232, 301, 379, 466, ...
%e 1, 10, 46, 136, 307, 586, 1000, 1576, 2341, 3322, 4546, ...
%e 1, 13, 79, 307, 886, 2086, 4258, 7834, 13327, 21331, 32521, ...
%t Table[Hypergeometric2F1[-n, -n, 1, 3], {n, 0, 21}] (* _Arkadiusz Wesolowski_, Aug 13 2012 *)
%o (PARI) a(n)=sum(k=0,n,binomial(n,k)^2*3^k)
%o (PARI) a(n)=if(n<0, 0, polcoeff((1+4*x+3*x^2)^n, n))
%o (PARI) /* as lattice paths: same as in A092566 but use */
%o steps=[[1,0], [0,1], [1,1], [1,1]]; /* note the double [1,1] */
%o \\ _Joerg Arndt_, Jul 01 2011
%o (PARI) a(n)=pollegendre(n,2)<<n \\ _Charles R Greathouse IV_, Mar 18 2017
%o (Haskell)
%o a069835 n = a081577 (2 * n) n -- _Reinhard Zumkeller_, Mar 16 2014
%o (GAP) List([0..25],n->Sum([0..n],k->Binomial(n,k)^2*3^k)); # _Muniru A Asiru_, Jul 29 2018
%Y Cf. A001850.
%K nonn,easy
%O 0,2
%A _Benoit Cloitre_, May 03 2002
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