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A276909 E.g.f. A(x) satisfies: Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)). 4
1, 0, 3, 0, 85, 0, 6111, 0, 872649, 0, 195062395, 0, 76208072733, 0, 12330526252695, 0, 125980697776559377, 0, -857710566759117989133, 0, 11428318296234746748941925, 0, -222333914273403535165432496561, 0, 6242434914385931957857138485252825, 0, -244888574110309970555770302512462694549, 0, 13082369513456349871152908238665975845490989, 0, -930879791318792717095933863751868808486774883065, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

It appears that a(6*k+5) = 1 (mod 3) for k>=0 with a(n) = 0 (mod 3) elsewhere.

Apart from signs, essentially the same as A276910.

E.g.f. A(x) equals the series reversion of the e.g.f. of A276908.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..301

FORMULA

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

(1) A( A(x)*exp(A(x)) ) = -LambertW(-x),

(2) A( A(x)*exp(-A(x)) ) = LambertW(x),

where LambertW( x*exp(x) ) = x.

(3) Series_Reversion( A( x*exp(x) ) ) = A( x*exp(-x) ).

EXAMPLE

E.g.f.: 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! - 222333914273403535165432496561*x^23/23! + 6242434914385931957857138485252825*x^25/25! +...

such that Series_Reversion( A(x)*exp(A(x)) ) = A(x)*exp(-A(x)).

RELATED SERIES.

A(x)*exp(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! + 63300353418240*x^15/15! + 1386349221087856*x^16/16! + 149879079531401472*x^17/17! +...+ A276911(n)*x^n/n! +...

exp(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! + 574020572798641*x^16/16! + 133507199865641216*x^17/17! +...+ A276912(n)*x^n/n! +...

Also, A( A(x)*exp(A(x)) ) = -LambertW(-x), which begins:

A( A(x)*exp(A(x)) ) = x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 625*x^5/5! + 7776*x^6/6! + 117649*x^7/7! + 2097152*x^8/8! +...+ n^(n-1)*x^n/n! +...

PROG

(PARI) {a(n) = my(A=x +x*O(x^n));

for(i=1, n, A = A + (x - subst(A*exp(A), x, A*exp(-A)))/2); n!*polcoeff(A, n)}

for(n=1, 30, print1(a(n), ", "))

(PARI) {a(n) = my(V=[1], A=x); for(i=1, n\2+1, V = concat(V, [0, 0]); A = sum(m=1, #V, V[m]*x^m/m!) +x*O(x^#V); V[#V] = -(#V)!/2 * polcoeff( subst( A*exp(A), x, A*exp(-A) ), #V) ); V[n]}

for(n=1, 30, print1(a(n), ", "))

CROSSREFS

Cf. A276910, A276911, A276912, A276913, A179270, A276908.

Sequence in context: A296621 A120953 A009784 * A276910 A013245 A013238

Adjacent sequences: A276906 A276907 A276908 * A276910 A276911 A276912

KEYWORD

sign

AUTHOR

Paul D. Hanna, Sep 26 2016

STATUS

approved

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Last modified November 29 12:29 EST 2022. Contains 358427 sequences. (Running on oeis4.)