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%I #70 Apr 19 2024 11:30:48
%S 1,7,73,847,10321,129367,1651609,21360031,278905249,3668760487,
%T 48543499753,645382441711,8614382884849,115367108888311,
%U 1549456900170553,20861640747345727,281483386791966529,3805228005705102151,51527535767904810889,698796718936034430607
%N P_n(7), where P_n is n-th Legendre polynomial; also, a(n) = central coefficient of (1 + 7*x + 12*x^2)^n.
%C More generally, given fixed parameters b and c, we have the identities:
%C (1) a(n) = Sum_{k=0..n} binomial(n,k)^2 * b^k * c^(n-k);
%C (2) a(n) = [x^n] (1 + (b+c)*x + b*c*x^2)^n;
%C (3) g.f.: 1/sqrt(1 - 2*(b+c)*x + (b-c)^2*x^2);
%C (4) Sum_{n>=1} a(n)*x^n/n = log(G(x)) where G(x) = 1 + (b+c)*x*G(x) + b*c*x^2*G(x)^2.
%C Number of directed 2-D walks of length 2n starting at (0,0) and ending on the X-axis using steps NE, SE, NE, SW and avoiding NE followed by SE. - _David Scambler_, Jun 24 2013
%H Michael De Vlieger, <a href="/A084768/b084768.txt">Table of n, a(n) for n = 0..875</a>
%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 G. Levy, <a href="http://purl.flvc.org/fsu/fd/FSU_migr_etd-3099">Solutions of second order recurrence equations</a> (2010) PhD Thesis, Florida State University, page 3.
%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 - 14*x + x^2).
%F Also a(n) = (n+1)-th term of the binomial transform of 1/(1-3x)^(n+1).
%F a(n) = Sum_{k=0..n} 3^k*C(n,k)*C(n+k,k). - _Benoit Cloitre_, Apr 13 2004
%F E.g.f.: exp(7*x)*Bessel_I(0, 2*sqrt(12)*x). - _Paul Barry_, May 25 2005
%F D-finite with recurrence: n*a(n) + 7*(1-2*n)*a(n-1) + (n-1)*a(n-2) = 0. - _R. J. Mathar_, Sep 27 2012
%F a(n) = Sum_{k=0..n} C(n,k)^2 * 3^k * 4^(n-k). - _Paul D. Hanna_, Sep 28 2012
%F a(n) ~ (7+4*sqrt(3))^(n+1/2)/(2*3^(1/4)*sqrt(2*Pi*n)). - _Vaclav Kotesovec_, Jul 31 2013
%F a(n) = hypergeom([-n, n+1], [1], -3). - _Peter Luschny_, May 23 2014
%F a(n)^2 = Sum_{k=0..n} 12^k * C(2*k, k)^2 * C(n+k, n-k) = A243944(n). - _Paul D. Hanna_, Aug 18 2014
%F From _Peter Bala_, Apr 17 2024: (Start)
%F a(n) = (1/4)*(1/3)^n*Sum_{k >= n} binomial(k, n)^2*(3/4)^k.
%F a(n) = (1/4)^(n+1)*hypergeom([n+1, n+1], [1], 3/4).
%F a(n) = [x^n] ((1 + x)*(4 + 3*x))^n = [x^n] ((1 + 3*x)*(1 + 4*x))^n.
%F a(n) = (3^n)*hypergeom([-n, -n], [1], 4/3) = (4^n)*hypergeom([-n, -n], [1], 3/4).
%F The Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.
%F a(n) = (-1)^n * Sum_{k = 0..n} (-4)^k*binomial(2*k, k)*binomial(n+k, n-k).
%F G.f: Sum_{n >= 0} (3^n)*binomial(2*n, n)*x^n/(1 - x)^(2*n+1) = 1 + 7*x + 73*x^2 + 847^x^3 + .... (End)
%t Table[LegendreP[n, 7], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 31 2013 *)
%o (PARI) for(n=0,30,print1(subst(pollegendre(n),x,7)","))
%o (PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*3^k*4^(n-k))} \\ _Paul D. Hanna_, Sep 28 2012
%o for(n=0, 20, print1(a(n), ", "))
%o (PARI) /* From a(n)^2 = A243944(n) (_Paul D. Hanna_, Aug 18 2014): */
%o {a(n) = sqrtint( sum(k=0, n, 12^k * binomial(2*k, k)^2 * binomial(n+k, n-k) ) )}
%o for(n=0, 20, print1(a(n), ", "))
%o (Magma) [Evaluate(LegendrePolynomial(n),7): n in [0..40]]; // _G. C. Greubel_, May 17 2023
%o (SageMath) [gen_legendre_P(n,0,7) for n in range(41)] # _G. C. Greubel_, May 17 2023
%Y Sequences of the form LegendreP(n, 2*m+1): A000012 (m=0), A001850 (m=1), A006442 (m=2), this sequence (m=3), A084769 (m=4).
%Y Cf. A084774, A243944 (a(n)^2).
%K nonn,easy,changed
%O 0,2
%A _Paul D. Hanna_, Jun 03 2003
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