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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. 5
1, 7, 73, 847, 10321, 129367, 1651609, 21360031, 278905249, 3668760487, 48543499753, 645382441711, 8614382884849, 115367108888311, 1549456900170553, 20861640747345727, 281483386791966529, 3805228005705102151, 51527535767904810889, 698796718936034430607 (list; graph; refs; listen; history; text; internal format)
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, László Szalay, Diagonal Sums in the Pascal Pyramid, II: Applications, J. Int. Seq., Vol. 22 (2019), Article 19.3.5.

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

Conjecture: 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

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), ", "))

CROSSREFS

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

Sequence in context: A071060 A092444 A099141 * A106651 A114429 A124547

Adjacent sequences:  A084765 A084766 A084767 * A084769 A084770 A084771

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jun 03 2003

STATUS

approved

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Last modified October 14 08:54 EDT 2019. Contains 327995 sequences. (Running on oeis4.)