A084768 P_n(7), where P_n is n-th Legendre polynomial; also, a(n) = central coefficient of (1 + 7*x + 12*x^2)^n.

1, 7, 73, 847, 10321, 129367, 1651609, 21360031, 278905249, 3668760487, 48543499753, 645382441711, 8614382884849, 115367108888311, 1549456900170553, 20861640747345727, 281483386791966529, 3805228005705102151, 51527535767904810889, 698796718936034430607

Offset

0,2

Comments

More generally, given fixed parameters b and c, we have the identities:

(1) a(n) = Sum_{k=0..n} binomial(n,k)^2 * b^k * c^(n-k);

(2) a(n) = [x^n] (1 + (b+c)*x + b*c*x^2)^n;

(3) g.f.: 1/sqrt(1 - 2*(b+c)*x + (b-c)^2*x^2);

(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.

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

Links

Michael De Vlieger, Table of n, a(n) for n = 0..875

Hacène Belbachir, Abdelghani Mehdaoui, and László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

G. Levy, Solutions of second order recurrence equations (2010) PhD Thesis, Florida State University, page 3.

Tony D. Noe, On the Divisibility of Generalized Central Trinomial Coefficients, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.7.

Formula

G.f.: 1/sqrt(1 - 14*x + x^2).

Also a(n) = (n+1)-th term of the binomial transform of 1/(1-3x)^(n+1).

a(n) = Sum_{k=0..n} 3^k*C(n,k)*C(n+k,k). - Benoit Cloitre, Apr 13 2004

E.g.f.: exp(7*x)*Bessel_I(0, 2*sqrt(12)*x). - Paul Barry, May 25 2005

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

a(n) = Sum_{k=0..n} C(n,k)^2 * 3^k * 4^(n-k). - Paul D. Hanna, Sep 28 2012

a(n) ~ (7+4*sqrt(3))^(n+1/2)/(2*3^(1/4)*sqrt(2*Pi*n)). - Vaclav Kotesovec, Jul 31 2013

a(n) = hypergeom([-n, n+1], [1], -3). - Peter Luschny, May 23 2014

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

From Peter Bala, Apr 17 2024: (Start)

a(n) = (1/4)*(1/3)^n*Sum_{k >= n} binomial(k, n)^2*(3/4)^k.

a(n) = (1/4)^(n+1)*hypergeom([n+1, n+1], [1], 3/4).

a(n) = [x^n] ((1 + x)*(4 + 3*x))^n = [x^n] ((1 + 3*x)*(1 + 4*x))^n.

a(n) = (3^n)*hypergeom([-n, -n], [1], 4/3) = (4^n)*hypergeom([-n, -n], [1], 3/4).

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.

a(n) = (-1)^n * Sum_{k = 0..n} (-4)^k*binomial(2*k, k)*binomial(n+k, n-k).

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)

Mathematica

Table[LegendreP[n, 7], {n, 0, 20}] (* Vaclav Kotesovec, Jul 31 2013 *)

Prog

(PARI) for(n=0, 30, print1(subst(pollegendre(n), x, 7)", "))
(PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*3^k*4^(n-k))} \\ Paul D. Hanna, Sep 28 2012
for(n=0, 20, print1(a(n), ", "))
(PARI) /* From a(n)^2 = A243944(n) (Paul D. Hanna, Aug 18 2014): */
{a(n) = sqrtint( sum(k=0, n, 12^k * binomial(2*k, k)^2 * binomial(n+k, n-k) ) )}
for(n=0, 20, print1(a(n), ", "))
(Magma) [Evaluate(LegendrePolynomial(n), 7): n in [0..40]]; // G. C. Greubel, May 17 2023
(SageMath) [gen_legendre_P(n, 0, 7) for n in range(41)] # G. C. Greubel, May 17 2023

Crossrefs

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).

Cf. A084774, A243944 (a(n)^2).

Sequence in context: A071060 A092444 A099141 * A357165 A357226 A106651

Adjacent sequences: A084765 A084766 A084767 * A084769 A084770 A084771

Keyword

nonn,easy,changed

Author

Paul D. Hanna, Jun 03 2003

Status

approved