A294784 E.g.f. A(x) satisfies: A'(x) = (1 + A(x)^2)^2/4.

1, 1, 2, 8, 46, 346, 3212, 35468, 453976, 6607936, 107781992, 1947158168, 38592660016, 832595731696, 19422479520992, 487137028505408, 13072025077208416, 373697069074031776, 11338183238037941312, 363881995144694554688, 12316073980019762824576, 438441199984650577010176, 16376568508223695174746752, 640396538780869661656846208, 26164698834332206196492375296, 1114866540340266230645081994496

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..400 (terms 0..300 from Paul D. Hanna)

Formula

E.g.f. A(x) satisfies:

(1) A(x) = d/dx sin( Series_Reversion( x + cos(x) - 1 ) ).

(2) A(x) = d/dx Series_Reversion( asin(x) + sqrt(1-x^2) - 1 ).

(3) A(x) = d/dx Series_Reversion( Integral sqrt(1-x^2)/(1+x) dx ).

(4) A(x) = sqrt(1 - B(x)^2) / (1 - B(x)) where B(x) = Integral A(x) dx.

(5) Integral A(x) dx = (A(x)^2 - 1)/(A(x)^2 + 1).

(6) A(x) = (A(x) + 1/A(x))/2 * ( 1 + Integral A(x) dx ).

(7) exp( Integral (A(x) + 1/A(x))/2 dx ) = 1 + Integral A(x) dx.

(8) A(x) = 1 + Integral (1 + A(x)^2)^2/4 dx.

a(n) ~ c * (2/(Pi-2))^n * n^(n-1/6) / exp(n), where c = 1.2415... - Vaclav Kotesovec, Nov 11 2017

Example

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 8*x^3/3! + 46*x^4/4! + 346*x^5/5! + 3212*x^6/6! + 35468*x^7/7! + 453976*x^8/8! + 6607936*x^9/9! + 107781992*x^10/10! + 1947158168*x^11/11! + 38592660016*x^12/12! + 832595731696*x^13/13! + 19422479520992*x^14/14! + 487137028505408*x^15/15! +...
such that A'(x) = (1 + A(x)^2)^2/4.
RELATED SERIES.
Series_Reversion( Integral sqrt(1-x^2)/(1+x) dx ) = x + x^2/2! + 2*x^3/3! + 8*x^4/4! + 46*x^5/5! + 346*x^6/6! + 3212*x^7/7! + 35468*x^8/8! +...
which equals Integral A(x) dx.
(A(x)^2 - 1)/(A(x)^2 + 1) = x + x^2/2! + 2*x^3/3! + 8*x^4/4! + 46*x^5/5! + 346*x^6/6! + 3212*x^7/7! + 35468*x^8/8! +...
which equals Integral A(x) dx.
Note that asin( Integral A(x) dx ) = Series_Reversion(x + cos(x) - 1), the e.g.f. of A200317.
A(x)^2 = 1 + 2*x + 6*x^2/2! + 28*x^3/3! + 180*x^4/4! + 1472*x^5/5! + 14616*x^6/6! + 170728*x^7/7! + 2293320*x^8/8! + 34822592*x^9/9! + 589761216*x^10/10! +...
(A(x) - 1/A(x))/2 = x + x^2/2! + 5*x^3/3! + 26*x^4/4! + 196*x^5/5! + 1786*x^6/6! + 19550*x^7/7! + 248156*x^8/8! + 3588916*x^9/9! + 58220416*x^10/10! +...
(A(x) + 1/A(x))/2 = 1 + x^2/2! + 3*x^3/3! + 20*x^4/4! + 150*x^5/5! + 1426*x^6/6! + 15918*x^7/7! + 205820*x^8/8! + 3019020*x^9/9! + 49561576*x^10/10! +...
where (A(x) - 1/A(x))/2  =  (A(x) + 1/A(x))/2 * Integral A(x) dx.

Prog

(PARI) {a(n) = my(A=1); A = deriv( sin( serreverse( x + cos(x +x^2*O(x^n)) - 1 ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = deriv( serreverse( intformal( sqrt( (1-x)/(1+x +x*O(x^n)) ) ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); for(i=1, n+1, A = 1 + intformal( (1 + A^2)^2/4 +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A294785, A200317.

Sequence in context: A229559 A128085 A052801 * A180390 A300696 A074599

Adjacent sequences: A294781 A294782 A294783 * A294785 A294786 A294787

Keyword

nonn

Author

Paul D. Hanna, Nov 09 2017

Status

approved