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 A274277 E.g.f. A(x) satisfies: A( A(x)^2 ) = x^2 * exp(-2*x). 1
 1, -2, 6, -40, 320, -2976, 35392, -538112, 9931392, -211790080, 5059784576, -132643057152, 3761875287040, -114501941915648, 3725395402721280, -129324055589257216, 4786638435256696832, -188785468724361560064, 7922155381738193944576, -352740315643746941665280, 16603695476218208847691776, -822951583413551750366298112, 42792449844854211313594597376, -2327246576567999111735900897280, 132052357036729088907927420928000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Table of n, a(n) for n=1..25. FORMULA E.g.f. equals the series reversion of the e.g.f. of A274275. Given e.g.f. A(x), let B(x) satisfy A(B(x)) = x, then (1) A(x) = sqrt( B( x^2*exp(-2*x) ) ). (2) A(x^2) = B(x)^2 * exp(-2*B(x)). EXAMPLE E.g.f.: A(x) = x - 2*x^2/2! + 6*x^3/3! - 40*x^4/4! + 320*x^5/5! - 2976*x^6/6! + 35392*x^7/7! - 538112*x^8/8! + 9931392*x^9/9! - 211790080*x^10/10! + 5059784576*x^11/11! - 132643057152*x^12/12! + 3761875287040*x^13/13! - 114501941915648*x^14/14! + 3725395402721280*x^15/15! - 129324055589257216*x^16/16! +... where A( A(x)^2 ) = x^2 * exp(-2*x). RELATED SERIES. Let B(x) be the series reversion of the e.g.f. A(x), which begins B(x) = x + 2*x^2/2! + 6*x^3/3! + 40*x^4/4! + 400*x^5/5! + 4656*x^6/6! + 62944*x^7/7! + 1046144*x^8/8! + 20274048*x^9/9! + 438238720*x^10/10! + 10529132416*x^11/11! + 280439144448*x^12/12! + 8185848206848*x^13/13! + 259202608222208*x^14/14! +...+ A274275(n)*x^n/n! +... then A(x) = sqrt( B( x^2*exp(-2*x) ) ) and A(x^2) = B(x)^2 * exp(-2*B(x)). A(x)^2 = 2*x^2/2! - 12*x^3/3! + 72*x^4/4! - 640*x^5/5! + 6960*x^6/6! - 85344*x^7/7! + 1226624*x^8/8! - 21007872*x^9/9! + 422254080*x^10/10! - 9724042240*x^11/11! + 250998494208*x^12/12! +... where A(x)^2 = B( x^2*exp(-2*x) ) such that B(A(x)) = x. PROG (PARI) /* From A(x) = sqrt( B( x^2*exp(-2*x) ) ) where A(B(x)) = x */ {a(n) = my(A=x, B=x); for(i=1, n, B = serreverse(A +x*O(x^n)); A = sqrt( subst(B, x, x^2*exp(-2*x +x*O(x^n))) ) ); n!*polcoeff(A, n)} for(n=1, 30, print1(a(n), ", ")) (PARI) /* As the series reversion of the e.g.f. of A274275 */ {a(n) = my(B=x); for(i=1, n, B = serreverse( sqrt( subst(B, x, x^2*exp(-2*x +x*O(x^n))) ) ) ); n!*polcoeff(serreverse(B), n)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A274275. Sequence in context: A056787 A098852 A274482 * A120592 A277476 A277483 Adjacent sequences: A274274 A274275 A274276 * A274278 A274279 A274280 KEYWORD sign AUTHOR Paul D. Hanna, Jun 18 2016 STATUS approved

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Last modified September 16 22:04 EDT 2024. Contains 375979 sequences. (Running on oeis4.)