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!)
A179330 E.g.f. satisfies: A(x) = (1+x)/(1+3*x) * A(x*(1+x)^2). 2
0, 2, -6, 42, -468, 7080, -133128, 2938824, -73169568, 1997384832, -58814501760, 1868053207680, -65311214042880, 2585560450337280, -115344597684718080, 5424254194395456000, -244310147229735014400, 10256126830544041574400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..17.

FORMULA

E.g.f. A=A(x) satisfies:

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

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

. G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...; G001764(x) = g.f. of A001764;

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

INVERSION FORMULA:

More generally, if A(x) = A(G(x)) * G(x)/(x*G'(x)) with G(0)=0, G'(0)=1,

then G(x) can be obtained from A=A(x) by the series:

. G(x)/x = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F).

ITERATION FORMULA:

Let G_n(x) denote the n-th iteration of G(x) = x*(1+x)^2, and A=A(x), then:

. A(x) = A(G_n(x)) * G_n(x)/(x*G_n'(x)) for all n;

. G_n(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! +... where Dx(F) = d/dx(x*F).

...

MATRIX LOG OF RIORDAN ARRAY (G(x)/x, G(x)) where G(x) = x*(1+x)^2:

. k*A(x) = e.g.f. of column k of the matrix log of triangle A116088 for k>=0.

EXAMPLE

E.g.f.: A(x) = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! - 133128*x^6/6! + 2938824*x^7/7! - 73169568*x^8/8! + 1997384832*x^9/9! - 58814501760*x^10/10! + 1868053207680*x^11/11! - 65311214042880*x^12/12! +...

...

A(x*(1+x)^2) = 2*x + 2*x^2/2! - 18*x^3/3! + 108*x^4/4! - 480*x^5/5! - 2808*x^6/6! + 162792*x^7/7! - 3940128*x^8/8! + 57267648*x^9/9! + 534366720*x^10/10! - 78703384320*x^11/11! + 2883142045440*x^12/12! +...

...

where A(x*(1+x)^2) = (1+3*x)/(1+x) * A(x).

...

Related expansions begin:

. A = 2*x - 6*x^2/2! + 42*x^3/3! - 468*x^4/4! + 7080*x^5/5! +...

. A*Dx(A)/2! = 8*x^2/2! - 90*x^3/3! + 1332*x^4/4! - 25200*x^5/5! +...

. A*Dx(A*Dx(A))/3! = 48*x^3/3! - 1248*x^4/4! + 32760*x^5/5! -+...

. A*Dx(A*Dx(A*Dx(A)))/4! = 384*x^4/4! - 18480*x^5/5! + 770400*x^6/6! -+...

. A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 3840*x^5/5! - 300672*x^6/6! +-...

...

Sums of which generate the square of the g.f. of A001764:

. G001764(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+...

. G001764(-x)^2 = 1 - 2*x + 7*x^2 - 30*x^3 + 143*x^4 - 728*x^5 +...+ A006013(n)*(-x)^n +...

...

The Riordan array ((1+x)^2, x*(1+x)^2) (cf. A116088) begins:

1;

2, 1;

1, 4, 1;

0, 6, 6, 1;

0, 4, 15, 8, 1;

0, 1, 20, 28, 10, 1;

0, 0, 15, 56, 45, 12, 1; ...

The matrix log of Riordan array ((1+x)^2, x*(1+x)^2) begins:

0;

2, 0;

-6/2!, 4, 0;

42/3!, -12/2!, 6, 0;

-468/4!, 84/3!, -18/2!, 8, 0;

7080/5!, -936/4!, 126/3!, -24/2!, 10, 0;

-133128/6!, 14160/5!, -1404/4!, 168/3!, -30/2!, 12, 0; ...

where the g.f. of the leftmost column equals the e.g.f. of this sequence.

PROG

(PARI) /* E.g.f. satisfies: A(x) = (1+x)/(1+3*x)*A(x*(1+x)^2): */

{a(n)=local(A=2*x, B); for(m=2, n, B=(1+x)/(1+3*x+O(x^(n+3)))*subst(A, x, x*(1+x)^2+O(x^(n+3))); A=A-polcoeff(B, m+1)*x^m/(m-1)/2); n!*polcoeff(A, n)}

(PARI) /* (1+x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */

{a(n)=local(A=0+sum(m=1, n-1, a(m)*x^m/m!), D=1, R=0); R=-((1+x)^2+x*O(x^n))+1+sum(m=1, n, (D=A*deriv(x*D+x*O(x^n)))/m!); -n!*polcoeff(R, n)}

(PARI) /* First column of the matrix log of triangle A116088: */

{a(n)=local(M=matrix(n+1, n+1, r, c, if(r>=c, polcoeff(((1+x)^2+x*O(x^n))^c, r-c))), LOG, ID=M^0); LOG=sum(m=1, n+1, -(ID-M)^m/m); n!*LOG[n+1, 1]}

CROSSREFS

Cf. A179331, variants: A179320, A179420.

Sequence in context: A258969 A161632 A115974 * A066864 A181737 A116896

Adjacent sequences:  A179327 A179328 A179329 * A179331 A179332 A179333

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jul 21 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 November 19 16:10 EST 2017. Contains 294936 sequences.