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FORMULA
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E.g.f. A=A(x) satisfies: x = 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).
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a(n) = -n*(n-2)!*Sum_{i=1..n-1} C(n-i+1,i+1)*a(n-i)/(n-i)! for n>1 with a(1)=1.
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a(n) = (-1)^(n-1)*n*A005119(n), where A005119 describes the infinitesimal generator of (x+x^2).
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Equals column 0 of A179198, the matrix log of triangle A030528, where A030528(n,k) = C(k,n-k); the g.f. of column k in A030528 is (x+x^2)^(k+1)/x.
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A179198(n,k) = (k+1)*a(n-k)/(n-1)! for n>0, k>=0, where A179198 = matrix log of triangle A030528.
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EXAMPLE
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E.g.f.: A(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 620*x^5/5! - 7536*x^6/6! + 109032*x^7/7! - 1809984*x^8/8! + 33562944*x^9/9! - 681799680*x^10/10! + 14980204800*x^11/11! - 354016189440*x^12/12! +...
E.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2) where:
. A(x+x^2) = x - 3*x^3/3! + 20*x^4/4! - 120*x^5/5! + 624*x^6/6! - 840*x^7/7! - 58752*x^8/8! + 1512000*x^9/9! - 25660800*x^10/10! +...
E.g.f. A = A(x) satisfies:
. x = 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) and expansions begin:
. A*Dx(A) = 4*x^2/2! - 30*x^3/3! + 288*x^4/4! - 3500*x^5/5! +-...
. A*Dx(A*Dx(A)) = 36*x^3/3! - 624*x^4/4! + 10680*x^5/5! -+...
. A*Dx(A*Dx(A*Dx(A))) = 576*x^4/4! - 18480*x^5/5! + 504000*x^6/6! -+...
. A*Dx(A*Dx(A*Dx(A*Dx(A)))) = 14400*x^5/5! - 751680*x^6/6! +-...
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PROG
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(PARI) /* E.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2): */
{a(n)=local(A=x, B); for(m=2, n, B=(1+x)/(1+2*x+O(x^(n+3)))*subst(A, x, x+x^2+O(x^(n+3))); A=A-polcoeff(B, m+1)*x^m/(m-1)); n!*polcoeff(A, n)}
(PARI) /* Recurrence (slow): */
{a(n)=if(n<1, 0, if(n==1, 1, -n*(n-2)!*sum(i=1, n-1, binomial(n-i+1, i+1)*a(n-i)/(n-i)!)))}
(PARI) /* x = A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...: */
{a(n)=local(A=x+sum(m=2, n-1, a(m)*x^m/m!), G=1, R=0); R=sum(m=1, n, (G=A*deriv(x*G+x*O(x^n)))/m!); if(n==1, 1, -n!*polcoeff(R, n))}
(PARI) /* As column 0 of the matrix log of triangle A030528: */
{a(n)=local(A030528=matrix(n+1, n+1, r, c, if(r>=c, binomial(c, r-c))), LOG, ID=A030528^0); LOG=sum(m=1, n+1, -(ID-A030528)^m/m); n!*LOG[n+1, 1]}
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