This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.