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 A193202 E.g.f. A(x) satisfies: A(A(x)) = x*A'(A(x)). 2
 1, 2, 0, 12, -160, 3240, -86688, 2922640, -119971584, 5847901920, -332122243200, 21653202377664, -1601381638172160, 133036354347921024, -12314128238585510400, 1261212911036957548800, -142082122642808666185728, 17514853400850824425213440, -2351847513553411263501035520, 342599734607249938595012582400 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Paul D. Hanna, Table of n, a(n) for n = 1..150 FORMULA E.g.f. satisfies: A( A(x)/A'(x) ) = x. E.g.f. satisfies: A(x) = Series_Reversion(-G(-x)) where G(x) = -A(-x)/A'(-x) is the e.g.f. of A179420 and satisfies: G(G(x)) = x*G'(x). The inverse function of A(x), A(x)/A'(x), equals the g.f. of column 0 of the matrix log of the Riordan array (A(x)/x, A(x)). Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x, then: _ A(x)/A'(x) = A_{n-1}(x) * A_n(x) / (x * d/dx A_n(x)) for all n. EXAMPLE E.g.f.: A(x) = x + 2*x^2/2! + 12*x^4/4! - 160*x^5/5! + 3240*x^6/6! - 86688*x^7/7! + 2922640*x^8/8! - 119971584*x^9/9! + 5847901920*x^10/10! +... RELATED EXPANSIONS. _ A(A(x)) = x + 4*x^2/2! + 12*x^3/3! + 48*x^4/4! + 40*x^5/5! + 2640*x^6/6! - 57456*x^7/7! + 2059904*x^8/8! - 85967136*x^9/9! + 4262310720*x^10/10! +... _ A'(A(x)) = 1 + 2*x + 4*x^2/2! + 12*x^3/3! + 8*x^4/4! + 440*x^5/5! - 8208*x^6/6! +... _ A(x)/A'(x) = x - 2*x^2/2! + 12*x^3/3! - 132*x^4/4! + 2200*x^5/5! - 50280*x^6/6! + 1482768*x^7/7! - 54171376*x^8/8! + 2381590944*x^9/9! - 123292821600*x^10/10! +... Higher order iterations begin: _ A_3(x) = x + 6*x^2/2! + 36*x^3/3! + 252*x^4/4! + 1800*x^5/5! + 16920*x^6/6! +... _ A_4(x) = x + 8*x^2/2! + 72*x^3/3! + 768*x^4/4! + 9200*x^5/5! + 126720*x^6/6! +... _ A_5(x) = x + 10*x^2/2! + 120*x^3/3! + 1740*x^4/4! + 29200*x^5/5! + 561000*x^6/6! +... Illustrate a main property of the iterations A_n(x) by: _ A(x)/A'(x) = A(x) * A(A(x)) / (x*d/dx A(A(x))); _ A(x)/A'(x) = A_2(x) * A_3(x) / (x*d/dx A_3(x)); _ A(x)/A'(x) = A_3(x) * A_4(x) / (x*d/dx A_4(x)); _ A(x)/A'(x) = A_4(x) * A_5(x) / (x*d/dx A_5(x)); ... which can be shown consistent by the chain rule of differentiation. PROG (PARI) {a(n)=local(A=x+x^2+sum(m=3, n-1, a(m)*x^m/m!)+x*O(x^n)); if(n<3, n!*polcoeff(A, n), n!*polcoeff(subst(A, x, A)-x*subst(A', x, A), n)/(n-2))} CROSSREFS Cf. A179420. Sequence in context: A013316 A013310 A058803 * A107700 A122688 A110685 Adjacent sequences:  A193199 A193200 A193201 * A193203 A193204 A193205 KEYWORD sign AUTHOR Paul D. Hanna, Jul 22 2011 STATUS approved

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