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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.)