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%I #50 May 20 2023 15:20:55
%S 1,9,169,3969,103041,2832489,80802121,2365752321,70611901441,
%T 2139090528969,65568745087209,2029206892664961,63300531617048961,
%U 1987912809986437161,62787371136571152009,1992942254830520803329,63531842302018973818881,2033004661359005674887561
%N Squares of the central Delannoy numbers: a(n) = A001850(n)^2.
%C In general, we have the binomial identity:
%C if b(n) = Sum_{k=0..n} t^k * C(2*k, k) * C(n+k, n-k), then b(n)^2 = Sum_{k=0..n} (t^2+t)^k * C(2*k, k)^2 * C(n+k, n-k), where the g.f. of b(n) is 1/sqrt(1 - (4*t+2)*x + x^2), and the g.f. of b(n)^2 is 1 / AGM(1-x, sqrt((1+x)^2 - (4*t+2)^2*x)), where AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.
%C Note that the g.f. of A001850 is 1/sqrt(1 - 6*x + x^2).
%C Limit_{n -> oo} a(n+1)/a(n) = (3 + 2*sqrt(2))^2 = 17 + 12*sqrt(2).
%C From _Gheorghe Coserea_, Jul 05 2016: (Start)
%C Diagonal of the rational function 1/(1 - x - y - z - x*y + x*z - y*z - x*y*z).
%C Annihilating differential operator: x*(x-1)*(x+1)*(x^2-34*x+1)*Dx^2 + (3*x^4-66*x^3-70*x^2+70*x-1)*Dx + x^3-7*x^2-35*x+9.
%C (End).
%C The sequence b(n) mentioned above is the sequence of shifted Legendre polynomials P(n,2*t + 1) (see A063007). See Zudilin for a g.f. for the sequence b(n)^2. - _Peter Bala_, Mar 02 2017
%H Seiichi Manyama, <a href="/A243949/b243949.txt">Table of n, a(n) for n = 0..500</a>
%H A. Bostan, S. Boukraa, J.-M. Maillard, and J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>
%H W. Zudilin, <a href="http://arxiv.org/abs/1210.2493">A generating function of the squares of Legendre polynomials</a>, arXiv:1210.2493v2 [math.CA], 2012.
%F G.f.: 1 / AGM(1-x, sqrt(1-34*x+x^2)). - _Paul D. Hanna_, Aug 30 2014
%F a(n) = Sum_{k=0..n} 2^k * C(2*k, k)^2 * C(n+k, n-k).
%F a(n)^(1/2) = Sum_{k=0..n} C(2*k, k) * C(n+k, n-k).
%F Recurrence: n^2*(2*n-3)*a(n) = (2*n-1)*(35*n^2 - 70*n + 26)*a(n-1) - (2*n-3)*(35*n^2 - 70*n + 26)*a(n-2) + (n-2)^2*(2*n-1)*a(n-3). - _Vaclav Kotesovec_, Aug 18 2014
%F a(n) ~ (4 + 3*sqrt(2)) * (3 + 2*sqrt(2))^(2*n) / (8*Pi*n). - _Vaclav Kotesovec_, Aug 18 2014
%F From _Gheorghe Coserea_, Jul 05 2016: (Start)
%F G.f.: hypergeom([1/12, 5/12],[1],27648*x^4*(x^2-34*x+1)*(x-1)^2/(1-36*x+134*x^2-36*x^3+x^4)^3)/(1-36*x+134*x^2-36*x^3+x^4)^(1/4).
%F 0 = x*(x-1)*(x+1)*(x^2-34*x+1)*y'' + (3*x^4-66*x^3-70*x^2+70*x-1)*y' + (x^3-7*x^2-35*x+9)*y, where y is g.f.
%F (End)
%F a(n) = Sum_{k = 0..n} 4^k*binomial(n+k,2*k)^2*binomial(2*k,k). - _Peter Bala_, Mar 02 2017
%F a(n) = hypergeom([1/2, -n, n + 1], [1, 1], -8). - _Peter Luschny_, Mar 14 2018
%F G.f.: Sum_{n >= 0} (2^n)*binomial(2*n,n)^2 *x^n/(1-x)^(2*n+1). - _Peter Bala_, Feb 07 2022
%e G.f.: A(x) = 1 + 9*x + 169*x^2 + 3969*x^3 + 103041*x^4 + 2832489*x^5 +...
%t Table[Sum[2^k *Binomial[2*k, k]^2 *Binomial[n+k, n-k], {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Aug 18 2014 *)
%t a[n_]:= HypergeometricPFQ[{1/2, -n, n+1}, {1, 1}, -8];
%t Table[a[n], {n, 0, 17}] (* _Peter Luschny_, Mar 14 2018 *)
%t LegendreP[Range[0, 30], 3]^2 (* _G. C. Greubel_, May 17 2023 *)
%o (PARI) {a(n) = sum(k=0, n, 2^k * binomial(2*k, k)^2 * binomial(n+k, n-k) )}
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) {a(n)=polcoeff( 1 / agm(1-x, sqrt((1+x)^2 - 36*x +x*O(x^n))), n)}
%o for(n=0, 20, print1(a(n), ", "))
%o (Python)
%o from math import comb
%o def A243949(n): return sum(comb(n,k)*comb(n+k,k) for k in range(n+1))**2 # _Chai Wah Wu_, Mar 23 2023
%o (Magma) [Evaluate(LegendrePolynomial(n), 3)^2 : n in [0..40]]; // _G. C. Greubel_, May 17 2023
%o (SageMath) [gen_legendre_P(n,0,3)^2 for n in range(41)] # _G. C. Greubel_, May 17 2023
%Y Sequences of the form LegendreP(n, 2*m+1)^2: A000012 (m=0), this sequence (m=1), A243943 (m=2), A243944 (m=3), A243007 (m=4).
%Y Related to diagonal of rational functions: A268545 - A268555.
%Y Cf. A001850, A243945, A245925.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Aug 17 2014
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