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
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! +...
-
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
-
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 *)
-
{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)}
A360989
E.g.f. satisfies A(x) = exp(x / A(-x)^2).
Original entry on oeis.org
1, 1, 5, 1, -231, 81, 55453, -40431, -30313231, 33477985, 29630916981, -43713004191, -45378051616631, 83666428734513, 100216964952070541, -221570666935625999, -301515678925659598623, 777062158771833364929, 1185517627245415533666277
Offset: 0
A360990
E.g.f. satisfies A(x) = exp(x / A(-x)^3).
Original entry on oeis.org
1, 1, 7, -8, -827, 2896, 452179, -2511872, -560237303, 4254259456, 1237434920191, -11907540107264, -4275828959720435, 49800209789734912, 21288959122755516235, -290981680034059649024, -144324916601232035246831, 2264121148389579474141184
Offset: 0
-
A360990 := proc(n)
add((-3*n+3*k+1)^(k-1)*(3*k)^(n-k)*binomial(n,k),k=0..n) ;
end proc:
seq(A360990(n),n=0..60) ; # R. J. Mathar, Mar 12 2023
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a(n) = sum(k=0, n, (-3*n+3*k+1)^(k-1)*(3*k)^(n-k)*binomial(n, k));
A385687
E.g.f. A(x) satisfies A(x) = exp( x*((A(x) + A(-x))/2)^2 ).
Original entry on oeis.org
1, 1, 1, 7, 25, 341, 2161, 44115, 404209, 11010025, 132273601, 4508793983, 67085545033, 2747071330173, 48765277295281, 2331905267846731, 48106649137922017, 2631174441142423505, 61862217319644572161, 3809106344377237185399, 100542158725584301036921
Offset: 0
-
terms = 21; A[] = 1; Do[A[x] = Exp[x*((A[x] + A[-x])/2)^2] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x]Range[0,terms-1]! (* Stefano Spezia, Jul 07 2025 *)
A385140
E.g.f. A(x) satisfies A(x) = exp(2*x*A(-x)^(1/2)).
Original entry on oeis.org
1, 2, 0, -22, -16, 1042, 1792, -116758, -330496, 24101090, 96518144, -7976308118, -41609056256, 3875582805746, 25008143335424, -2601876338050582, -20048671462064128, 2308957345471798978, 20711293319504723968, -2618684079639256157974, -26823633677081126109184
Offset: 0
Showing 1-6 of 6 results.