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A246467 G.f.:  1 / AGM(1-5*x, sqrt((1-x)*(1-25*x))). 4
1, 9, 121, 2025, 38025, 762129, 15912121, 341621289, 7484845225, 166549691025, 3751508008161, 85341068948529, 1957289174870121, 45199191579030225, 1049893021288265625, 24510327614556266025, 574726636455361317225, 13528549573868347823025, 319541915502909478890625 (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

Table of n, a(n) for n=0..18.

FORMULA

a(n) = A026375(n)^2 = [Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)]^2.

G.f.:  1 / AGM((1-x)*(1+5*x), (1+x)*(1-5*x)) = Sum_{n>=0} a(n)*x^(2*n).

EXAMPLE

G.f.: A(x) = 1 + 9*x + 121*x^2 + 2025*x^3 + 38025*x^4 + 762129*x^5 +...

where the square-root of the terms yields A026375:

[1, 3, 11, 45, 195, 873, 3989, 18483, 86515, 408105, ...]

the g.f. of which is 1/sqrt((1-x)*(1-5*x)).

PROG

(PARI) {a(n)=polcoeff( 1 / agm(1-5*x, sqrt((1-x)*(1-25*x) +x*O(x^n))), n)}

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

(PARI) {a(n)=sum(k=0, n, binomial(n, k)*binomial(2*k, k))^2}

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

CROSSREFS

Cf. A026375, A168597, A246876, A246906, A248167.

Sequence in context: A138978 A046184 A084769 * A202835 A050353 A112941

Adjacent sequences:  A246464 A246465 A246466 * A246468 A246469 A246470

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 06 2014

STATUS

approved

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Last modified October 16 12:35 EDT 2018. Contains 316263 sequences. (Running on oeis4.)