A138740
G.f. satisfies A(x) = A(A(x)) - x^2 with A(0)=0.
Original entry on oeis.org
1, 1, -2, 9, -56, 420, -3572, 33328, -334354, 3559310, -39838760, 465743720, -5658983108, 71191948512, -924554859776, 12365546196641, -169995491295312, 2398380272232272, -34680290150700800, 513390937937217088, -7773229533145403728
Offset: 1
G.f.: A(x) = x + x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-..;
A(A(x)) = x + 2*x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
The g.f. satisfies:
A(x) = x + x^2*exp((x-A(x))/x + [d/dx (x-A(x))^2/x]/2! + [d^2/dx^2 (x-A(x))^3/x]/3! + [d^3/dx^3 (x-A(x))^4/x]/4! +...)^2.
Higher order iterations of A=A(x) may be expressed in terms of A and x:
A(A(x)) = A + x^2 ;
A(A(A(x))) = (A + A^2) + x^2 ;
A(A(A(A(x)))) = (A + 2*A^2) + (1 + 2*A)*x^2 + x^4 ;
A(A(A(A(A(x))))) = (A + 3*A^2 + 2*A^3 + A^4) + (1 + 4*A + 2*A^2)*x^2 + 2*x^4 ;
A(A(A(A(A(A(x)))))) = (A + 4*A^2 + 6*A^3 + 5*A^4) + (1 + 6*A + 10*A^2 + 8*A^3)*x^2 + (3 + 6*A + 8*A^2)*x^4 + (2 + 4*A)*x^6 + x^8 ;
A(A(A(A(A(A(A(x))))))) = (A + 5*A^2 + 12*A^3 + 18*A^4 + 14*A^5 + 10*A^6 + 4*A^7 + A^8) + (1 + 8*A + 24*A^2 + 40*A^3 + 30*A^4 + 16*A^5 + 4*A^6)*x^2 + (4 + 18*A + 40*A^2 + 24*A^3 + 8*A^4)*x^4 + (6 + 20*A + 8*A^2)*x^6 + 5*x^8 .
The sums of coefficients in the above expansions form A000278: [1,1,2,3,7,16,65,321,4546,107587,20773703,...].
Let G(x) = Series_Reversion(A(x)) = g.f. of A139702, then
G(x) = x - x^2 + 4*x^3 - 24*x^4 + 178*x^5 - 1512*x^6 +-...
G(x)^2 = x^2 - 2*x^3 + 9*x^4 - 56*x^5 + 420*x^6 - 3572*x^7 +-...
so that G(x)^2 = A(x) - x and G(x + G(x)^2) = x.
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{a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n,A=A-polcoeff(subst(A,x,A+x*O(x^i))-x^2,i)*x^i);polcoeff(A,n))}
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{a(n)=local(A=x); if(n<1, 0, for(i=1,n, A=serreverse(x + (A+x*O(x^n))^2)); polcoeff(serreverse(A), n))}
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/* n-th Derivative: */
{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
/* G.f.: [Paul D. Hanna, Mar 24 2011] */
{a(n)=local(A=x+x^2+x*O(x^n)); for(i=1, n, A=x+x^2*exp(2*sum(m=0, n, Dx(m, (x-A)^(m+1)/x)/(m+1)!)+x*O(x^n))); polcoeff(A, n)}
A138913
G.f. A(x) satisfies: 4*A(x) = A(A(A(x))) + 3*x + x^2 with A(0)=0.
Original entry on oeis.org
1, 1, 6, 99, 2362, 70484, 2463460, 97309959, 4251047468, 202470323828, 10409697289888, 573563068625768, 33682595044746416, 2099111839596600644, 138339363094940014088, 9612941947359915802978, 702527738704990333954432
Offset: 1
G.f.: A(x) = x + x^2 + 6*x^3 + 99*x^4 + 2362*x^5 + 70484*x^6 +...
A(A(x)) = x + 2*x^2 + 14*x^3 + 229*x^4 + 5456*x^5 + 162710*x^6 +...
A(A(A(x))) = x + 3*x^2 + 24*x^3 + 396*x^4 + 9448*x^5 + 281936*x^6 +...
so that 4*A(x) = A(A(A(x))) + 3*x + x^2.
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{a(n)=local(A=x+x^2);if(n<1,0, for(i=3,n+1,A=A+polcoeff(subst(A,x,subst(A,x,A+x*O(x^i))),i)*x^i); polcoeff(A,n))}
A138914
G.f. A(x) satisfies: 5*A(x) = A(A(A(A(x)))) + 4*x + x^2 with A(0)=0.
Original entry on oeis.org
1, 1, 12, 390, 18304, 1071862, 73349996, 5661162666, 482252816998, 44704184452202, 4465265748489708, 477159108766899654, 54255973609630750372, 6536766146592886952548, 831617552461457925554152
Offset: 1
G.f.: A(x) = x + x^2 + 12*x^3 + 390*x^4 + 18304*x^5 + 1071862*x^6 +...
A(A(x)) = x + 2*x^2 + 26*x^3 + 841*x^4 + 39440*x^5 + 2308752*x^6 +...
A(A(A(x))) = x + 3*x^2 + 42*x^3 + 1359*x^4 + 63730*x^5 + 3730610*x^6 +...
A(A(A(A(x)))) = x + 4*x^2 + 60*x^3 + 1950*x^4 + 91520*x^5 + 5359310*x^6 +...
so that 5*A(x) = A(A(A(A(x)))) + 4*x + x^2.
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{a(n)=local(A=x+x^2,G);if(n<1,0,for(i=3,n+1,G=x; for(j=1,4,G=subst(A,x,G+x*O(x^i)));A=A+polcoeff(G,i)*x^i);polcoeff(A,n))}
A138915
G.f. A(x) satisfies: 6*A(x) = A(A(A(A(A(x))))) + 5*x + x^2 with A(0)=0.
Original entry on oeis.org
1, 1, 20, 1070, 82620, 7950630, 893138136, 113042205894, 15776443441194, 2393774318253534, 391021817774684352, 68276246115093735882, 12675272091572931300360, 2491402163326687657447940
Offset: 1
G.f.: A(x) = x + x^2 + 20*x^3 + 1070*x^4 + 82620*x^5 +...
A(A(x)) = x + 2*x^2 + 42*x^3 + 2241*x^4 + 172960*x^5 +...
A(A(A(x))) = x + 3*x^2 + 66*x^3 + 3519*x^4 + 271550*x^5 +...
A(A(A(A(x)))) = x + 4*x^2 + 92*x^3 + 4910*x^4 + 378944*x^5 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 120*x^3 + 6420*x^4 + 495720*x^5 +...
so that 6*A(x) = A(A(A(A(A(x))))) + 5*x + x^2.
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{a(n)=local(A=x+x^2,G);if(n<1,0,for(i=3,n+1,G=x; for(j=1,5,G=subst(A,x,G+x*O(x^i)));A=A+polcoeff(G,i)*x^i);polcoeff(A,n))}
A138916
G.f. A(x) satisfies: 7*A(x) = A(A(A(A(A(A(x)))))) + 6*x + x^2 with A(0)=0.
Original entry on oeis.org
1, 1, 30, 2385, 273560, 39078970, 6512700536, 1222156339336, 252751878117712, 56798072762849412, 13733835430565197700, 3548014267149570778764, 974073193845291808779496, 283008950620416071533339000
Offset: 1
G.f.: A(x) = x + x^2 + 30*x^3 + 2385*x^4 + 273560*x^5 +...
A(A(x)) = x + 2*x^2 + 62*x^3 + 4921*x^4 + 564280*x^5 +...
A(A(A(x))) = x + 3*x^2 + 96*x^3 + 7614*x^4 + 872950*x^5 +...
A(A(A(A(x)))) = x + 4*x^2 + 132*x^3 + 10470*x^4 + 1200384*x^5 +...
A(A(A(A(A(x))))) = x + 5*x^2 + 170*x^3 + 13495*x^4 + 1547420*x^5 +...
A(A(A(A(A(A(x)))))) = x + 6*x^2 + 210*x^3 + 16695*x^4 + 1914920*x^5 +...
so that 7*A(x) = A(A(A(A(A(A(x)))))) + 6*x + x^2.
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{a(n)=local(A=x+x^2,G);if(n<1,0,for(i=3,n+1,G=x; for(j=1,6,G=subst(A,x,G+x*O(x^i)));A=A+polcoeff(G,i)*x^i);polcoeff(A,n))}
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