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%I #5 Sep 04 2015 19:56:01
%S 0,3,-6,33,-288,3240,-41328,538776,-6772608,108502848,-3985200000,
%T 160423053120,-1094935449600,-334098799856640,6171528273039360,
%U 2583214608931891200,-150276838897808179200,-27910708844766496358400,3428432556793417713254400,462153194693572097337753600,-107869378361311784037187584000
%N E.g.f. satisfies: A(x) = A( x/(1-x)^3 ) * (1-x)/(1+2*x) with A(0)=0.
%F E.g.f. A = A(x) satisfies:
%F (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! +...
%F (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.
%F (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! +...
%F where Dx(F) = d/dx(x*F).
%e 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! +...
%e such that
%e 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! +...
%e where
%e A(x/(1-x)^3) = (1+2*x)/(1-x)*A(x).
%o (PARI) /* E.g.f. satisfies: A(x) = (1-x)/(1+2*x)*A(x/(1-x)^3): */
%o {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)}
%o for(n=0,20,print1(a(n),", "))
%o (PARI) /* 1/(1-x)^3 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */
%o {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)}
%o for(n=0,20,print1(a(n),", "))
%Y Cf. A261886, A179320, A179420, A179330.
%K sign
%O 0,2
%A _Paul D. Hanna_, Sep 04 2015
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