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A179497 E.g.f. satisfies: A(A(x))^2 = A(x)^2 * A'(x). 4

%I

%S 1,2,18,312,8240,297000,13705776,776778688,52511234688,4143702216960,

%T 375403993060800,38537107042934016,4435139176244554752,

%U 567238312617468850176,80029364113424328422400

%N E.g.f. satisfies: A(A(x))^2 = A(x)^2 * A'(x).

%H Paul D. Hanna, <a href="/A179497/b179497.txt">Table of n, a(n), n=1..100.</a>

%F E.g.f. A(x) satisfies: A(x)^2/x equals the e.g.f. of column 0 in the matrix log of the Riordan array (A(x)/x, A(x)).

%F Let A_n(x) denote the n-th iteration of e.g.f. A(x) with A_0(x)=x,

%F then L=A(x)^2/x satisfies the series:

%F . A(x)/x = 1 + L + L*Dx(L)/2! + L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! +...

%F . A_{-1}(x)/x = 1 - L + L*Dx(L)/2! - L*Dx(L*Dx(L))/3! + L*Dx(L*Dx(L*Dx(L)))/4! -+...

%F . A_n(x)/x = 1 + n*L + n^2*L*Dx(L)/2! + n^3*L*Dx(L*Dx(L))/3! + n^4*L*Dx(L*Dx(L*Dx(L)))/4! +...

%F where Dx(F) = d/dx(x*F).

%F Further, we have:

%F . [A_{n+1}(x)]^2 = A(x)^2*A_n'(x)

%F which holds for all n.

%e E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +..

%e Related expansions:

%e . A(x)/x = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +..

%e . A(x)^2/x = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +..

%e . A'(x) = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! +...

%e . A(A(x)) = x + 4*x^2/2! + 48*x^3/3! + 1008*x^4/4! + 30880*x^5/5! +...

%e . A(A(x))^2 = 2*x^2/2! + 24*x^3/3! + 480*x^4/4! + 13920*x^5/5! +...

%e Illustrate a main property of the iterations A_n(x) of A(x) by:

%e . [A_3(x)]^2 = A(x)^2 * A_2'(x);

%e . [A_4(x)]^2 = A(x)^2 * A_3'(x);

%e . [A_5(x)]^2 = A(x)^2 * A_4'(x); ...

%e which can be shown to hold by the chain rule of differentiation.

%e ...

%e The RIORDAN ARRAY (A(x)/x, A(x)) begins:

%e . 1;

%e . 1, 1;

%e . 6/2!, 2, 1;

%e . 78/3!, 14/2!, 3, 1;

%e . 1648/4!, 192/3!, 24/2!, 4, 1;

%e . 49500/5!, 4136/4!, 348/3!, 36/2!, 5, 1;

%e . 1957968/6!, 124840/5!, 7680/4!, 552/3!, 50/2!, 6, 1;

%e . 97097336/7!, 4928256/6!, 233940/5!, 12520/4!, 810/3!, 66/2!, 7, 1; ...

%e where the e.g.f. of column k = [A(x)/x]^(k+1) for k>=0.

%e ...

%e The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:

%e . 0;

%e . 1, 0;

%e . 4/2!, 2, 0;

%e . 42/3!, 8/2!, 3, 0;

%e . 768/4!, 84/3!, 12/2!, 4, 0;

%e . 20680/5!, 1536/4!, 126/3!, 16/2!, 5, 0;

%e . 749040/6!, 41360/5!, 2304/4!, 168/3!, 20/2!, 6, 0;

%e . 34497792/7!, 1498080/6!, 62040/5!, 3072/4!, 210/3!, 24/2!, 7, 0; ...

%e where the e.g.f. of column k = (k+1)*A(x)^2/x for k>=0.

%o (PARI) {a(n)=local(A=x+1*x^2+sum(m=3,n-1,a(m)*x^m/m!)+O(x^(n+3)));if(n<3,n!*polcoeff(A,n),n!*polcoeff(subst(A,x,A)^2/A^2,n-1)/(n-2))}

%Y Cf. A179498, A179499, A179420.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 31 2010

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Last modified December 11 15:43 EST 2017. Contains 295905 sequences.