A084768 a(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.

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

Offset: 0

Author

Paul D. Hanna, Jun 03 2003

Keywords

• nonn
• easy
• changed

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.

Crossrefs

Column k=3 of A335333.

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).

Programs

Magma

[Evaluate(LegendrePolynomial(n),7): n in [0..40]]; // G. C. Greubel, May 17 2023

Mathematica

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

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), ", "))

SageMath

[gen_legendre_P(n,0,7) for n in range(41)] # G. C. Greubel, May 17 2023

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)

a(n) = (-1)^n * Sum_{k=0..n} (1/14)^(n-2*k) * binomial(-1/2,k) * binomial(k,n-k). - Seiichi Manyama, Aug 28 2025

a(n) = Sum_{k=0..floor(n/2)} 12^k * 7^(n-2*k) * binomial(n,2*k) * binomial(2*k,k). - Seiichi Manyama, Aug 30 2025