OFFSET
0,2
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
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jul 21 2010
STATUS
approved