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A179320 E.g.f. satisfies: A(x) = A( x/(1-x)^2 ) * (1-x)/(1+x) with A(0)=0. 4
0, 2, -2, 6, -28, 160, -936, 4536, -20448, 627264, -19699200, 43908480, 17788273920, -211715112960, -41219197125120, 1301670191808000, 160057006683033600, -10037518414724505600, -1007362871616478003200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

J. Cigler, Some results and conjectures about a class of q-polynomials with simple moments, 2014; http://homepage.univie.ac.at/Johann.Cigler/preprints/q-pol.pdf

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..200

FORMULA

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

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

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

(3) (1-x)^2/(1-3*x+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! +...

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:

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! +...

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

E.g.f. A(x) forms column 0 of A179321, the matrix log of triangle A078812, where A078812(n,k) = C(n+k+1,n-k); the g.f. of column k in A078812 is [x/(1-x)^2]^(k+1)/x.

A179321(n,k) = (k+1)*a(n-k)/(n-k)! for n>0, k>=0, where A179321 = matrix log of triangle A078812.

...

a(n) = (-1)^(n-1)*2*A027614(n), where A027614 is related to Clebsch-Gordan formulas.

EXAMPLE

E.g.f.: A(x) = 2*x - 2*x^2/2! + 6*x^3/3! - 28*x^4/4! + 160*x^5/5! - 936*x^6/6! + 4536*x^7/7! - 20448*x^8/8! + 627264*x^9/9! - 19699200*x^10/10! + 43908480*x^11/11! + 17788273920*x^12/12! -+...

A(x/(1-x)^2) = 2*x + 6*x^2/2! + 18*x^3/3! + 68*x^4/4! + 360*x^5/5! + 2184*x^6/6! + 13272*x^7/7! + 122016*x^8/8! + 1541376*x^9/9! + 1987200*x^10/10! - 150923520*x^11/11! + 16504093440*x^12/12! +...

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

...

Related expansions begin:

. A = 2*x - 2*x^2/2! + 6*x^3/3! - 28*x^4/4! + 160*x^5/5! +...

. A*Dx(A)/2! = 8*x^2/2! - 30*x^3/3! + 180*x^4/4! - 1400*x^5/5! +...

. A*Dx(A*Dx(A))/3! = 48*x^3/3! - 416*x^4/4! + 4280*x^5/5! +...

. A*Dx(A*Dx(A*Dx(A)))/4! = 384*x^4/4! - 6160*x^5/5! + 98400*x^6/6! -+...

. A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 3840*x^5/5! - 100224*x^6/6! +-...

where Catalan(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! +-... = 1 - 2*x + 5*x^2 - 14*x^3 + 42*x^4 +...+ A000108(n)*(-x)^n +...

PROG

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

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

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

(PARI) /* As column 0 of the matrix log of triangle A078812: */

{a(n)=local(A078812=matrix(n+1, n+1, r, c, if(r>=c, binomial(r+c-1, r-c))), LOG, ID=A078812^0); LOG=sum(m=1, n+1, -(ID-A078812)^m/m); n!*LOG[n+1, 1]}

CROSSREFS

Cf. A261885, A179321, A027614, A078812, variant: A179199.

Sequence in context: A076726 A032272 A214446 * A004304 A108800 A270487

Adjacent sequences:  A179317 A179318 A179319 * A179321 A179322 A179323

KEYWORD

sign

AUTHOR

Paul D. Hanna, Jul 11 2010

STATUS

approved

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Last modified November 23 09:38 EST 2017. Contains 295115 sequences.