A248167 G.f.: 1 / AGM(1-33*x, sqrt((1-9*x)*(1-121*x))).

1, 49, 3249, 261121, 23512801, 2266426449, 228110356881, 23642146057761, 2502698427758529, 269194720423487089, 29319711378381802609, 3225762406810715071041, 357859427246543331576481, 39977637030683399494792849, 4492572407488016429783217489

Offset

0,2

Comments

In general, the g.f. of the squares of coefficients in g.f. 1/sqrt((1-p*x)*(1-q*x)) is given by

1/AGM(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x))) = Sum_{n>=0} x^n*[Sum_{k=0..n} p^(n-k)*((q-p)/4)^k*C(n,k)*C(2*k,k)]^2,

and consists of integer coefficients when 4|(q-p).

Here AGM(x,y) = AGM((x+y)/2,sqrt(x*y)) is the arithmetic-geometric mean.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..481

Formula

a(n) = A248168(n)^2 = [ Sum_{k=0..n} 3^(n-k)*2^k * C(n,k) * C(2*k,k) ]^2.

G.f.: 1 / AGM((1-3*x)*(1+11*x), (1+3*x)*(1-11*x)) = Sum_{n>=0} a(n)*x^(2*n).

a(n) ~ 11^(2*n + 1) / (8*Pi*n). - Vaclav Kotesovec, Sep 27 2019

Example

G.f.: A(x) = 1 + 49*x + 3249*x^2 + 261121*x^3 + 23512801*x^4 +...
where the square-root of the terms yields A248168:
[1, 7, 57, 511, 4849, 47607, 477609, 4862319, 50026977, ...],
the g.f. of which is 1/sqrt((1-3*x)*(1-11*x)).

Mathematica

a[n_] := Sum[3^(n - k) * 2^k * Binomial[n, k] * Binomial[2k, k], {k, 0, n} ]^2; Array[a, 17, 0] (* Amiram Eldar, Dec 11 2018 *)

Prog

(PARI) {a(n, p=3, q=11)=polcoeff( 1 / agm(1-p*q*x, sqrt((1-p^2*x)*(1-q^2*x) +x*O(x^n))), n) }
for(n=0, 20, print1(a(n, 3, 11), ", "))
(PARI) {a(n, p=3, q=11)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }
for(n=0, 20, print1(a(n, 3, 11), ", "))
(PARI) {a(n, p=3, q=11)=sum(k=0, n, p^(n-k)*((q-p)/4)^k*binomial(n, k)*binomial(2*k, k))^2 }
for(n=0, 20, print1(a(n, 3, 11), ", "))

Crossrefs

Cf. A248168, A246923, A246467, A246906, A084771.

Sequence in context: A123841 A014773 A132539 * A218590 A372811 A218204

Adjacent sequences: A248164 A248165 A248166 * A248168 A248169 A248170

Keyword

nonn

Author

Paul D. Hanna, Oct 03 2014

Status

approved