login
This site is supported by donations to The OEIS Foundation.

 

Logo

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!)
A179498 E.g.f. satisfies: A(x) = A(x*A(x))^2 - x*A'(x). 2
1, 1, 6, 78, 1648, 49500, 1957968, 97097336, 5834581632, 414370221696, 34127635732800, 3211425586911168, 341164552018811904, 40517022329819203584, 5335290940894955228160, 773591071307555130451200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..15.

FORMULA

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

E.g.f.: A(x) = G(x)/x where G(x) = e.g.f. of A179497.

Let G_n(x) denote the n-th iteration of x*A(x) with G_0(x)=x, then

. [G_{n+1}(x)/x]^2 = A(x)^2*G_n'(x) for all n,

and L=x*A(x)^2 satisfies the series:

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

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

. G_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).

EXAMPLE

E.g.f.: A(x) = 1 + x + 6*x^2/2! + 78*x^3/3! + 1648*x^4/4! + 49500*x^5/5! +...

Related expansions:

. x*A(x) = x + 2*x^2/2! + 18*x^3/3! + 312*x^4/4! + 8240*x^5/5! +...

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

. x*A'(x) = x + 12*x^2/2! + 234*x^3/3! + 6592*x^4/4! + 247500*x^5/5! +...

. A(x*A(x)) = 1 + x + 8*x^2/2! + 132*x^3/3! + 3400*x^4/4! + 120940*x^5/5! +...

. A(x*A(x))^2 = 1 + 2*x + 18*x^2/2! + 312*x^3/3! + 8240*x^4/4! + 297000*x^5/5! +...

Illustrate the iterations G_n(x) of G(x) = x*A(x) by:

. [G_3(x)/x]^2 = A(x)^2 * G_2'(x);

. [G_4(x)/x]^2 = A(x)^2 * G_3'(x);

. [G_5(x)/x]^2 = A(x)^2 * G_4'(x); ...

which can be shown 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 g.f. of column k = A(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 g.f. of column k = (k+1)*x*A(x)^2 for k>=0.

PROG

(PARI) {a(n)=local(A=1+x+sum(m=2, n-1, a(m)*x^m/m!)+x*O(x^(n+5))); if(n<2, n!*polcoeff(A, n), n!*polcoeff(subst(A, x, x*A)^2, n)/(n-1))}

CROSSREFS

Cf. A179497, A179499, A179421.

Sequence in context: A229044 A049209 A162656 * A177556 A219435 A219135

Adjacent sequences:  A179495 A179496 A179497 * A179499 A179500 A179501

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 31 2010

STATUS

approved

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.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 12 16:18 EST 2017. Contains 295939 sequences.