A277612 E.g.f. satisfies: A(x - sin(x)^2) = x + sin(x)^2.

1, 4, 24, 224, 2880, 47104, 935424, 21853184, 587089920, 17829167104, 603915485184, 22571950997504, 922735222456320, 40954197741666304, 1961183862263906304, 100787274348058640384, 5532701353887903252480, 323102311113161602760704, 20000832981651983154806784, 1308180577070098190616756224, 90146906116103034082689024000, 6527896185206802934447948693504

Offset: 1

Author

Paul D. Hanna, Nov 06 2016

Keywords

• nonn

Examples

			E.g.f.: A(x) = x + 4*x^2/2! + 24*x^3/3! + 224*x^4/4! + 2880*x^5/5! + 47104*x^6/6! + 935424*x^7/7! + 21853184*x^8/8! + 587089920*x^9/9! + 17829167104*x^10/10! +...
such that A(x - sin(x)^2) = x + sin(x)^2.
RELATED SERIES.
A(x - sin(x)^2) = x + 2*x^2/2! - 8*x^4/4! + 32*x^6/6! - 128*x^8/8! + 512*x^10/10! - 2048*x^12/12! +...
which equals x + sin(x)^2.
cos(A(x) + x) = 1 - 4*x^2/2! - 24*x^3/3! - 224*x^4/4! - 2880*x^5/5! - 47104*x^6/6! +...
which equals 1+x - A(x).
		

Crossrefs

Cf. A143134.

Programs

PARI

{a(n) = my(A=x); for(i=1,21,A = subst(x + sin(x +x*O(x^n) )^2,x, serreverse(x - sin(x +x*O(x^n) )^2))); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

PARI

{a(n) = my(A=x); for(i=0,n, A = 1+x - cos(A + x +x*O(x^n))); n!*polcoeff(A,n)}
for(n=1,25,print1(a(n),", "))

Formula

G.f. A(x) also satisfies:

(1) A(x) = 1+x - cos(A(x) + x).

(2) A(x) = x + 2 * sin( (A(x) + x)/2 )^2.

(3) A(x) = -x + 2 * Series_Reversion(x - sin(x)^2).

a(n) = 2 * A143134(n) for n>1.