A276912 E.g.f.: A(x) satisfies: Series_Reversion( log(A(x)) * A(x) ) = log(A(x)) / A(x).

1, 1, 1, 4, 13, 116, 661, 8632, 70617, 1247248, 13329001, 285675776, 3782734693, 107823153088, 1685127882621, 28683829833856, 574020572798641, 133507199865641216, 2477747434090344913, -832289494713919714304, -16453576647394853560899, 11260772482520581109810176, 246622016098219255086463333, -219530418791080092679815129088, -5247252347909156791432867741559, 6177525915951437030555334153342976, 160073955175697692672876432040185401

Offset

0,4

Links

Paul D. Hanna, Table of n, a(n) for n = 0..300

Formula

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

(1) A( log(A(x)) / A(x) ) = x / LambertW(x).

(2) A( log(A(x)) * A(x) ) = LambertW(-x) / (-x).

(3) A(x) * A(-x) = 1.

Example

E.g.f.: A(x) = 1 + x + x^2/2! + 4*x^3/3! + 13*x^4/4! + 116*x^5/5! + 661*x^6/6! + 8632*x^7/7! + 70617*x^8/8! + 1247248*x^9/9! + 13329001*x^10/10! + 285675776*x^11/11! + 3782734693*x^12/12! + 107823153088*x^13/13! + 1685127882621*x^14/14! + 28683829833856*x^15/15! +...
such that Series_Reversion( log(A(x)) * A(x) ) = log(A(x)) / A(x).
RELATED SERIES.
The logarithm of the e.g.f. is an odd function:
log(A(x)) = x + 3*x^3/3! + 85*x^5/5! + 6111*x^7/7! + 872649*x^9/9! + 195062395*x^11/11! + 76208072733*x^13/13! + 12330526252695*x^15/15! + 125980697776559377*x^17/17! - 857710566759117989133*x^19/19! + 11428318296234746748941925*x^21/21! +...+ i^(n-1)*A276910(n)*x^n/n! +...
and thus A(x) = 1/A(-x).
log(A(x)) * A(x) = x + 2*x^2/2! + 6*x^3/3! + 28*x^4/4! + 180*x^5/5! + 1446*x^6/6! + 13888*x^7/7! + 156472*x^8/8! + 2034000*x^9/9! + 29724490*x^10/10! + 476806176*x^11/11! + 8502508884*x^12/12! + 174802753216*x^13/13! + 3768345692398*x^14/14! +...+ A276911(n)*x^n/n! +...
log(A(x)) / A(x) = x - 2*x^2/2! + 6*x^3/3! - 28*x^4/4! + 180*x^5/5! - 1446*x^6/6! + 13888*x^7/7! - 156472*x^8/8! +...+ (-1)^(n-1)*A276911(n)*x^n/n! +...
RELATION TO LambertW(x):
A( log(A(x)) * A(x) ) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! +...+ (n+1)^(n-1)*x^n/n! +...
which equals LambertW(-x) / (-x).
A( log(A(x)) / A(x) ) = 1 + x - x^2/2! + 4*x^3/3! - 27*x^4/4! + 256*x^5/5! - 3125*x^6/6! + 46656*x^7/7! +...+ (n-1)^(n-1)*(-x)^n/n! +...
which equals x / LambertW(x).

Prog

(PARI) {a(n) = my(A=1+x, L); for(i=1, n, L = log(A +x*O(x^n)); A = exp( sqrt( L*A* serreverse(L*A) ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A276910, A276911.

Sequence in context: A261785 A331013 A362283 * A045886 A015460 A121813

Adjacent sequences: A276909 A276910 A276911 * A276913 A276914 A276915

Keyword

sign

Author

Paul D. Hanna, Sep 24 2016

Status

approved