A261885 E.g.f. satisfies: A(x) = A( x/(1-x)^3 ) * (1-x)/(1+2*x) with A(0)=0.

0, 3, -6, 33, -288, 3240, -41328, 538776, -6772608, 108502848, -3985200000, 160423053120, -1094935449600, -334098799856640, 6171528273039360, 2583214608931891200, -150276838897808179200, -27910708844766496358400, 3428432556793417713254400, 462153194693572097337753600, -107869378361311784037187584000

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

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

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

(2) G(-x)^3 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+..., where G(x) = 1 + x*G(x)^3 is the g.f. of A001764.

(3) (1-x)^6/((1-x)^3 - x)^3 = 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).

Example

E.g.f.: A(x) = 3*x - 6*x^2/2! + 33*x^3/3! - 288*x^4/4! + 3240*x^5/5! - 41328*x^6/6! + 538776*x^7/7! - 6772608*x^8/8! + 108502848*x^9/9! - 3985200000*x^10/10! +...
such that
A(x/(1-x)^3) = 3*x + 12*x^2/2! + 33*x^3/3! + 108*x^4/4! + 900*x^5/5! + 2952*x^6/6! - 19152*x^7/7! + 1694592*x^8/8! + 1847232*x^9/9! - 1796670720*x^10/10! +...
where
A(x/(1-x)^3) = (1+2*x)/(1-x)*A(x).

Prog

(PARI) /* E.g.f. satisfies: A(x) = (1-x)/(1+2*x)*A(x/(1-x)^3): */
{a(n)=local(A=3*x, B); for(m=2, n, B=(1-x)/(1+2*x+O(x^(n+3)))*subst(A, x, x/(1-x+O(x^(n+3)))^3); A=A-polcoeff(B, m+1)*x^m/(m-1)/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* 1/(1-x)^3 = 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))^3+1+sum(m=1, n, (D=A*deriv(x*D+x*O(x^n)))/m!); -n!*polcoeff(R, n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A261886, A179320, A179420, A179330.

Sequence in context: A184508 A101142 A298679 * A375448 A186750 A203715

Adjacent sequences: A261882 A261883 A261884 * A261886 A261887 A261888

Keyword

sign

Author

Paul D. Hanna, Sep 04 2015

Status

approved