A186284 Self-convolution square equals A127776.
1, 2, 48, 1704, 71490, 3291780, 160844160, 8189867280, 429832053840, 23088359467040, 1263134996327680, 70138971602098560, 3942799810867610280, 223942062435751452240, 12831882367225056387840, 740872398293620831990080
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 48*x^2 + 1704*x^3 + 71490*x^4 + 3291780*x^5 +...
Related expansions.
The g.f. of A127776 equals A(x)^2:
A(x)^2 = 1 + 4*x + 100*x^2 + 3600*x^3 + 152100*x^4 + 7033104*x^5 +...+ A004981(n)^2*x^n +...
The g.f. of A002897 equals A(x)^4:
A(x)^4 = 1 + 8*x + 216*x^2 + 8000*x^3 + 343000*x^4 + 16003008*x^5 +...+ A000984(n)^3*x^n +...
The g.f. of A004981 begins:
1/(1-8*x)^(1/4) = 1 + 2*x + 10*x^2 + 60*x^3 + 390*x^4 + 2652*x^5 +...
where A004981(n) = (2^n/n!)*Product_{k=0..n-1} (4k + 1).
The g.f. of A000984 begins:
1/(1-4*x)^(1/2) = 1 + 2*x + 6*x^2 + 20*x^3 + 70*x^4 + 252*x^5 +...
where A000984(n) = (2n)!/(n!)^2 forms the central binomial coefficients.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..500
Programs
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Mathematica
nmax = 20; CoefficientList[Series[Sqrt[Hypergeometric2F1[ 1/4, 1/4, 1, 64*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *) -
PARI
{a(n)=local(A004981=1/(1-8*x+x*O(x^n))^(1/4),A=sum(m=0,n,polcoeff(A004981,m)^2*x^m+x*O(x^n))^(1/2));polcoeff(A,n)} -
PARI
{a(n)=local(A000984=1/(1-4*x+x*O(x^n))^(1/2),A=sum(m=0,n,polcoeff(A000984,m)^3*x^m+x*O(x^n))^(1/4));polcoeff(A,n)}
Formula
Self-convolution 4th power equals A002897.
G.f.: sqrt( K(k)/(Pi/2) ) in powers of (kk'/4)^2, where K(k) is complete elliptic integral of first kind evaluated at modulus k. [From a formula by Michael Somos in A002897]
G.f.: sqrt( 1/AGM(1, (1-16x)^(1/2)) ) in powers of x(1-16x) where AGM() is the arithmetic-geometric mean. [From a formula by Michael Somos in A004981]
a(n) ~ Pi^(3/4) * 2^(6*n - 1/2) / (Gamma(1/4)^3 * n^(3/2)). - Vaclav Kotesovec, Apr 10 2018