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

1, 2, 18, 312, 8240, 297000, 13705776, 776778688, 52511234688, 4143702216960, 375403993060800, 38537107042934016, 4435139176244554752, 567238312617468850176, 80029364113424328422400

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n), n=1..100.

Formula

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)).

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

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

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

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

. 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! +...

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

Further, we have:

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

which holds for all n.

Example

E.g.f.: A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +..
Related expansions:
. A(x)/x = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +..
. A(x)^2/x = x + 4*x^2/2! + 42*x^3/3! + 768*x^4/4! + 20680*x^5/5! +..
. A'(x) = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! +...
. A(A(x)) = x + 4*x^2/2! + 48*x^3/3! + 1008*x^4/4! + 30880*x^5/5! +...
. A(A(x))^2 = 2*x^2/2! + 24*x^3/3! + 480*x^4/4! + 13920*x^5/5! +...
Illustrate a main property of the iterations A_n(x) of A(x) by:
. [A_3(x)]^2 = A(x)^2 * A_2'(x);
. [A_4(x)]^2 = A(x)^2 * A_3'(x);
. [A_5(x)]^2 = A(x)^2 * A_4'(x); ...
which can be shown to hold by the chain rule of differentiation.
...
The RIORDAN ARRAY (A(x)/x, A(x)) begins:
. 1;
. 1, 1;
. 6/2!, 2, 1;
. 78/3!, 14/2!, 3, 1;
. 1648/4!, 192/3!, 24/2!, 4, 1;
. 49500/5!, 4136/4!, 348/3!, 36/2!, 5, 1;
. 1957968/6!, 124840/5!, 7680/4!, 552/3!, 50/2!, 6, 1;
. 97097336/7!, 4928256/6!, 233940/5!, 12520/4!, 810/3!, 66/2!, 7, 1; ...
where the e.g.f. of column k = [A(x)/x]^(k+1) for k>=0.
...
The MATRIX LOG of the above Riordan array (A(x)/x, A(x)) begins:
. 0;
. 1, 0;
. 4/2!, 2, 0;
. 42/3!, 8/2!, 3, 0;
. 768/4!, 84/3!, 12/2!, 4, 0;
. 20680/5!, 1536/4!, 126/3!, 16/2!, 5, 0;
. 749040/6!, 41360/5!, 2304/4!, 168/3!, 20/2!, 6, 0;
. 34497792/7!, 1498080/6!, 62040/5!, 3072/4!, 210/3!, 24/2!, 7, 0; ...
where the e.g.f. of column k = (k+1)*A(x)^2/x for k>=0.

Prog

(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))}

Crossrefs

Cf. A179498, A179499, A179420.

Sequence in context: A370941 A350366 A370056 * A296837 A326270 A227325

Adjacent sequences: A179494 A179495 A179496 * A179498 A179499 A179500

Keyword

nonn

Author

Paul D. Hanna, Jul 31 2010

Status

approved