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%I #9 Aug 24 2017 09:24:39
%S 1,1,1,1,5,14,29,73,229,671,1840,5415,16983,52547,161420,511039,
%T 1655598,5372395,17527912,58076084,194676024,656160449,2227549164,
%U 7635624954,26380508479,91696805060,320866223000,1130833326852,4010720214072,14306769257286
%N G.f. satisfies: A(x) = x^3 - x + Series_Reversion(x - x*A(x)).
%C Compare the g.f. to a g.f. G(x) of A088714 (offset 1), which satisfies:
%C G(x) = Series_Reversion(x - x*G(x)),
%C G(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*G(x)^n/n!, and
%C G(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*G(x)^n/n! ).
%H Vaclav Kotesovec, <a href="/A212919/b212919.txt">Table of n, a(n) for n = 3..200</a>
%F G.f. A(x) also satisfies:
%F (1) A(x) = x^3 + Sum_{n>=1} d^(n-1)/dx^(n-1) x^n*A(x)^n/n!.
%F (2) A(x) = x^3 - x + x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(n-1)*A(x)^n/n! ).
%e G.f.: A(x) = x^3 + x^4 + x^5 + x^6 + 5*x^7 + 14*x^8 + 29*x^9 + 73*x^10 +...
%e The series reversion of x - x*A(x) begins:
%e x + x^4 + x^5 + x^6 + 5*x^7 + 14*x^8 + 29*x^9 + 73*x^10 + 229*x^11 +...
%e which equals x - x^3 + A(x).
%e The g.f. satisfies:
%e A(x) = x^3 + x*A(x) + d/dx x^2*A(x)^2/2! + d^2/dx^2 x^3*A(x)^3/3! + d^3/dx^3 x^4*A(x)^4/4! +...
%e log(1-x^2 + A(x)/x) = A(x) + d/dx x*A(x)^2/2! + d^2/dx^2 x^2*A(x)^3/3! + d^3/dx^3 x^3*A(x)^4/4! +...
%e Related expansions:
%e d/dx x^2*A(x)^2/2! = 4*x^7 + 9*x^8 + 15*x^9 + 22*x^10 + 78*x^11 + 260*x^12 +...
%e d^2/dx^2 x^3*A(x)^3/3! = 22*x^10 + 78*x^11 + 182*x^12 + 350*x^13 + 1080*x^14 +...
%e d^3/dx^3 x^4*A(x)^4/4! = 140*x^13 + 680*x^14 + 2040*x^15 + 4845*x^16 +...
%e d^4/dx^4 x^5*A(x)^5/5! = 969*x^16 + 5985*x^17 + 21945*x^18 + 61985*x^19 +...
%e ...
%e d^(n-1)/dx^(n-1) x^n*A(x)^n/n! = A002293(n)*x^(3*n+1) +...
%o (PARI) {a(n)=local(A=x^3); for(i=1, n, A=x^3-x+serreverse(x-x*A +x*O(x^n))); polcoeff(A, n)}
%o for(n=3, 40, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
%o {a(n)=local(A=x^3); for(i=1, n, A=x^3+sum(m=1, n, Dx(m-1, x^m*A^m/m!)+x*O(x^n))); polcoeff(A, n)}
%o for(n=3, 40, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
%o {a(n)=local(A=x^3); for(i=1, n, A=x^3-x+x*exp(sum(m=1, n, Dx(m-1, x^(m-1)*A^m/m!)+x*O(x^n)))); polcoeff(A, n)}
%o for(n=3, 40, print1(a(n), ", "))
%Y Cf. A088714, A212910.
%K nonn
%O 3,5
%A _Paul D. Hanna_, May 31 2012
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