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A248167 G.f.: 1 / AGM(1-33*x, sqrt((1-9*x)*(1-121*x))). 7

%I #17 Sep 27 2019 16:56:10

%S 1,49,3249,261121,23512801,2266426449,228110356881,23642146057761,

%T 2502698427758529,269194720423487089,29319711378381802609,

%U 3225762406810715071041,357859427246543331576481,39977637030683399494792849,4492572407488016429783217489

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

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

%C 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,

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

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

%H Seiichi Manyama, <a href="/A248167/b248167.txt">Table of n, a(n) for n = 0..481</a>

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

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

%F a(n) ~ 11^(2*n + 1) / (8*Pi*n). - _Vaclav Kotesovec_, Sep 27 2019

%e G.f.: A(x) = 1 + 49*x + 3249*x^2 + 261121*x^3 + 23512801*x^4 +...

%e where the square-root of the terms yields A248168:

%e [1, 7, 57, 511, 4849, 47607, 477609, 4862319, 50026977, ...],

%e the g.f. of which is 1/sqrt((1-3*x)*(1-11*x)).

%t 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 *)

%o (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) }

%o for(n=0, 20, print1(a(n,3,11), ", "))

%o (PARI) {a(n,p=3,q=11)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }

%o for(n=0, 20, print1(a(n,3,11), ", "))

%o (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 }

%o for(n=0, 20, print1(a(n,3,11), ", "))

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 03 2014

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)