A275764 E.g.f. satisfies: A(x) = exp(2*x) * A(-x).
1, 1, 2, 4, 16, 56, 426, 2262, 26944, 191536, 3126160, 27728240, 575175624, 6103078632, 153600146896, 1895624842048, 56097022625536, 789039958221824, 26841919568551488, 423728844983247552, 16289858574401789440, 285136754661527448832, 12223695878727911987200, 234939121837394575935488, 11111439664638562836316800, 232614372016075736439705216, 12030859273551523180503859456, 272479395898122444403210189312
Offset: 0
Keywords
Examples
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! +... and satisfies: A(x) = exp(2*x) * A(-x). RELATED SERIES. A(x) = G(x)^G(x) where 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! +... such that: G(x)^G(x) = exp(2*x) * G(-x)^G(-x). sqrt( A(x)*A(-x) ) = exp(-x) * A(x) = exp(x) * A(-x) where 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! +... 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! +... 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! +...
Crossrefs
Cf. A274377.
Programs
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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)} for(n=0, 40, print1(a(n), ", "))
Formula
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.