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%I #9 Sep 07 2016 13:05:53
%S 1,1,2,4,16,56,426,2262,26944,191536,3126160,27728240,575175624,
%T 6103078632,153600146896,1895624842048,56097022625536,789039958221824,
%U 26841919568551488,423728844983247552,16289858574401789440,285136754661527448832,12223695878727911987200,234939121837394575935488,11111439664638562836316800,232614372016075736439705216,12030859273551523180503859456,272479395898122444403210189312
%N E.g.f. satisfies: A(x) = exp(2*x) * A(-x).
%F E.g.f.: A(x) = G(x)^G(x), where G(x) = 1 + Series_Reversion( log( sqrt( (1+x)^(1+x) / (1-x)^(1-x) ) ) ) is the e.g.f. of A274377.
%e E.g.f.: A(x) = 1 + x + 2*x^2/2! + 4*x^3/3! + 16*x^4/4! + 56*x^5/5! + 426*x^6/6! + 2262*x^7/7! + 26944*x^8/8! + 191536*x^9/9! + 3126160*x^10/10! +...
%e and satisfies: A(x) = exp(2*x) * A(-x).
%e RELATED SERIES.
%e A(x) = G(x)^G(x) where
%e G(x) = 1 + x + x^3/3! + 16*x^5/5! + 736*x^7/7! + 67096*x^9/9! + 10163176*x^11/11! + 2306198896*x^13/13! + 732199108096*x^15/15! + 309860700130816*x^17/17! + 168568765338224896*x^19/19! +...+ A274377(n)*x^n/n! +...
%e such that: G(x)^G(x) = exp(2*x) * G(-x)^G(-x).
%e sqrt( A(x)*A(-x) ) = exp(-x) * A(x) = exp(x) * A(-x) where
%e sqrt( A(x)*A(-x) ) = 1 + x^2/2! + 9*x^4/4! + 275*x^6/6! + 18585*x^8/8! + 2230149*x^10/10! + 418527593*x^12/12! + 113225111103*x^14/14! + 41730188633073*x^16/16! +...
%e A(x)*A(-x) = 1 + 2*x^2/2! + 24*x^4/4! + 820*x^6/6! + 58240*x^8/8! + 7172448*x^10/10! + 1366904704*x^12/12! + 373500984064*x^14/14! + 138613162768896*x^16/16! +...
%e log(A(x)) = x + x^2/2! + 6*x^4/4! + 170*x^6/6! + 11200*x^8/8! + 1328304*x^10/10! + 247677584*x^12/12! + 66739336768*x^14/14! + 24532666253568*x^16/16! +...
%o (PARI) {a(n) = my(G,X = x + x^2*O(x^n)); G = 1 + serreverse( log( sqrt( (1+X)^(1+x)/(1-X)^(1-x) ) ) ); n!*polcoeff(G^G, n)}
%o for(n=0, 40, print1(a(n), ", "))
%Y Cf. A274377.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 26 2016
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