A248167 Expansion of 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, 507445676088537643607528136801

Offset: 0

Author

Paul D. Hanna, Oct 03 2014

Keywords

• nonn

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.

Examples

			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)).
		

Links

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

Crossrefs

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

Programs

Magma

m:=40;
A248168:= [n le 2 select 7^(n-1) else (7*(2*n-3)*Self(n-1) - 33*(n-2)*Self(n-2))/(n-1) : n in [1..m+2]];
A248167:= func< n | (A248168[n+1])^2 >;
[A248167(n): n in [0..m]]; // G. C. Greubel, May 31 2025

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 *)

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), ", "))

SageMath

@CachedFunction
def b(n): # b = A248168
     if (n<2): return 7^n
     else: return (7*(2*n-1)*b(n-1) - 33*(n-1)*b(n-2))//n
def A248167(n): return (b(n))^2
print([A248167(n) for n in range(41)]) # G. C. Greubel, May 31 2025

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