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%I #9 Jun 11 2024 17:08:59
%S 1,1,0,-5,0,196,0,-21440,0,4605736,0,-1636894280,0,869411900176,0,
%T -645115754969600,0,637400080589929216,0,-808996241179323833600,0,
%U 1282689609051390935443456,0,-2484567925086557616137108480,0,5773170916638182711440802000896,0,-15849498359717283169328377665597440,0,50754498157679863282024469922251431936,0,-187503919340846371804132353057069945159680
%N E.g.f. satisfies: A(x)^A(-x) = exp(2*x) * A(-x)^A(x).
%F E.g.f.: 1 + Series_Reversion( log( sqrt( (1+x)^(1-x) / (1-x)^(1+x) ) ) ).
%e E.g.f.: A(x) = 1 + x - 5*x^3/3! + 196*x^5/5! - 21440*x^7/7! + 4605736*x^9/9! - 1636894280*x^11/11! + 869411900176*x^13/13! - 645115754969600*x^15/15! + ...
%e such that A(x)^A(-x) = exp(2*x) * A(-x)^A(x).
%e RELATED SERIES.
%e A(x)^A(-x) = 1 + x - 2*x^2/2! - 8*x^3/3! + 56*x^4/4! + 336*x^5/5! - 4566*x^6/6! - 36826*x^7/7! + 771840*x^8/8! + 7854064*x^9/9! - 225103120*x^10/10! - 2770846704*x^11/11! + 101183120136*x^12/12! +...
%e Series_Reversion(A(x) - 1) = x + 5*x^3/6 + 9*x^5/20 + 13*x^7/42 + 17*x^9/72 + 21*x^11/110 + 25*x^13/156 + 29*x^15/210 +...+ (4*n+1)*x^(2*n+1)/(2*n*(2*n+1)) +...
%e which equals ( (1-x)*log(1+x) - (1+x)*log(1-x) )/2.
%o (PARI) {a(n) = my(A=1,X=x + x^2*O(x^n)); A = 1 + serreverse( log( sqrt( (1+X)^(1-x) / (1-X)^(1+x) ) ) ); n!*polcoeff(A, n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A274377.
%K sign
%O 0,4
%A _Paul D. Hanna_, Aug 24 2016
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