A360988
E.g.f. A(x) satisfies A(x) = exp(x * A(-x)^3).
Original entry on oeis.org
1, 1, -5, -44, 829, 14656, -488897, -13063616, 629051449, 22531502080, -1420908901469, -63859764079616, 4983153798630709, 269501734545522688, -25073583375908431769, -1585437525801020801024, 171326697778165116452977, 12401692280007001315999744
Offset: 0
A171193
G.f. A(x) satisfies A(x) = 1/(1 - x*A(2*x)^3).
Original entry on oeis.org
1, 1, 7, 109, 3207, 174581, 17929279, 3559607005, 1389312382199, 1075527698708485, 1658535837898129263, 5105026337441341642861, 31395991691829167745766311, 385982564381552315528268500501
Offset: 0
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nmax = 15; A[] = 0; Do[A[x] = 1/(1 - x*A[2*x]^3) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] (* Vaclav Kotesovec, Nov 03 2021 *)
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{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1/(1-x*subst(A, x, 2*x)^3) ); polcoeff(A, n)}
A168600
E.g.f. A(x) satisfies A(x) = exp( x*A(2*x)^2 ).
Original entry on oeis.org
1, 1, 9, 265, 19889, 3506801, 1417530745, 1302573091513, 2700478102745057, 12518436654808255585, 128568477648089286062441, 2900655737241126221237790185, 142677722979145454671155940121233, 15200178301599487957128451391538504145
Offset: 0
E.g.f: A(x) = 1 + x + 9*x^2/2! + 265*x^3/3! + 19889*x^4/4! +...
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F:= A -> A(x) - exp(x*A(2*x)^2):
Extend:= proc(ff)
local f1x, m, f2, S, R, i;
f1x:= ff(x); m:= degree(f1x,x);
f2:= unapply(ff(x) + add(a[i]*x^i,i=m+1..2*m+1),x);
S:= series(F(f2),x,2*m+2);
R:= solve(identity(convert(S,polynom),x),{seq(a[i],i=m+1..2*m+1)});
unapply(subs(R, f2(x)),x);
end proc:
g:= 1:
for iter from 1 to 5 do g:= Extend(g) od:
seq(coeff(g(x),x,j)*j!,j=0..31); # Robert Israel, Feb 22 2019
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nmax = 13; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x] - Exp[x A[2 x]^2] + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq][[1]], {n, 1, nmax}];
sol /. HoldPattern[a[n_] -> k_] :> Set[a[n], k n!];
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)
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{a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=exp(x*subst(A, x, 2*x)^2) ); n!*polcoeff(A, n)}
A385651
E.g.f. A(x) satisfies A(x) = exp( x*(A(x) + A(2*x))^3 ).
Original entry on oeis.org
1, 8, 640, 150272, 81879040, 97446821888, 252536538529792, 1441194498488532992, 18238881125752291459072, 511646632486244583515095040, 31662959021226253504069431721984, 4295217009165735294411016058313900032, 1268984197722535033624735101886101792489472
Offset: 0
Showing 1-4 of 4 results.