A186284 Self-convolution square equals A127776.

1, 2, 48, 1704, 71490, 3291780, 160844160, 8189867280, 429832053840, 23088359467040, 1263134996327680, 70138971602098560, 3942799810867610280, 223942062435751452240, 12831882367225056387840, 740872398293620831990080

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..500

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

Example

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.

Mathematica

nmax = 20; CoefficientList[Series[Sqrt[Hypergeometric2F1[ 1/4, 1/4, 1, 64*x]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2018 *)

Prog

(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)}

Crossrefs

Cf. A004981, A000984, A127776, A002897.

Sequence in context: A013517 A027509 A348322 * A119695 A119698 A112103

Adjacent sequences: A186281 A186282 A186283 * A186285 A186286 A186287

Keyword

nonn

Author

Paul D. Hanna, Feb 16 2011

Status

approved