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A246923 G.f.:  1 / AGM(1-9*x, sqrt((1-x)*(1-81*x))). 7
1, 25, 1089, 60025, 3690241, 241025625, 16359689025, 1140463805625, 81081830657025, 5852177325225625, 427465780890020929, 31528177440967935225, 2344153069158724611841, 175473167541934734763225, 13211212029033949825064769, 999630716942846408773325625 (list; graph; refs; listen; history; text; internal format)
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..525

FORMULA

a(n) = A084771(n)^2 = [Sum_{k=0..n} 2^k*C(n,k)*C(2*k,k)]^2.

G.f.:  1 / AGM((1-x)*(1+9*x), (1+x)*(1-9*x)) = Sum_{n>=0} a(n)*x^(2*n).

a(n) ~ 3^(4*n+2) / (8*Pi*n). - Vaclav Kotesovec, Sep 27 2019

EXAMPLE

G.f.: A(x) = 1 + 16*x + 324*x^2 + 7744*x^3 + 206116*x^4 + 5875776*x^5 +...

where the square-root of the terms yields A084771:

[1, 5, 33, 245, 1921, 15525, 127905, 1067925, ...],

the g.f. of which is 1/sqrt((1-x)*(1-9*x)).

MATHEMATICA

a[n_] := Sum[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=1, q=9)=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), ", "))

(PARI) {a(n, p=1, q=9)=polcoeff( 1 / sqrt((1-p*x)*(1-q*x) +x*O(x^n)), n)^2 }

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n, p=1, q=9)=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), ", "))

CROSSREFS

Cf. A246467, A248167, A246906, A084771.

Sequence in context: A167257 A012692 A193121 * A066852 A123204 A012508

Adjacent sequences:  A246920 A246921 A246922 * A246924 A246925 A246926

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 07 2014

STATUS

approved

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Last modified January 24 22:45 EST 2022. Contains 350565 sequences. (Running on oeis4.)