A232691 -id:A232691 - cp's OEIS frontend

A232690 E.g.f. satisfies: A(x) = exp( 1/A(x) * Integral A(x)^3 dx ).

1, 1, 2, 7, 33, 202, 1495, 13107, 132062, 1508629, 19227687, 270818542, 4173948097, 69906444393, 1263811926338, 24534217063999, 508951297964193, 11236656534791578, 263054502440239639, 6508910392250017611, 169727899004807970782, 4652123984505282141277, 133711980572082349859559

Offset: 0

Paul D. Hanna, Dec 06 2013

nonn

Note that G(x) = exp(1/G(x) * Integral G(x)^2 dx) has negative coefficients.

Compare e.g.f. to: B(x) = exp( 1/B(x) * Integral B(x) dx ) where B(y) = Bessel polynomial y_n(-1) (cf. A000806).

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 33*x^4/4! + 202*x^5/5! +...
Related expansions:
log(A(x)) = x + x^2/2! + 3*x^3/3! + 11*x^4/4! + 61*x^5/5! + 393*x^6/6! +...
Integral A(x)^3 dx = x + 3*x^2/2! + 12*x^3/3! + 63*x^4/4! + 411*x^5/5! +...
1/A(x) = 1 - x - x^3/3! - x^4/4! - 12*x^5/5! - 41*x^6/6! - 451*x^7/7! -...

Vaclav Kotesovec, Table of n, a(n) for n = 0..100

Cf. A232691, A232692.

seq(n! * coeff(series(sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)), x, n+1), x, n), n=0..20); # Vaclav Kotesovec, Jan 05 2014

CoefficientList[FullSimplify[Assuming[Element[x, Reals], Series[Sqrt[LambertW[-1,(4*x-3)*E^(-3)]/(4*x-3)], {x, 0, 20}]]], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 05 2014 *)

{a(n)=local(A=1+x);for(i=1,n,A=exp(1/A*intformal(A^3+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: sqrt(LambertW(-1,(4*x-3)*exp(-3))/(4*x-3)). - Vaclav Kotesovec, Jan 05 2014

Limit n->infinity (a(n)/n!)^(1/n) = 4/3. - Vaclav Kotesovec, Jan 05 2014

A232692 E.g.f. satisfies: A(x) = exp( 1/A(x)^3 * Integral A(x)^8 dx ).

Original entry on oeis.org

1, 1, 3, 24, 213, 3096, 46071, 967608, 20251809, 555747048, 15004870731, 508165972056, 16810393586733, 677183788645704, 26523956467895103, 1238567261126084856, 56056407696184372281, 2976966230117448265128, 152872356339113679491859, 9098430770913969095416728

Offset: 0

Paul D. Hanna, Dec 06 2013

nonn

Compare e.g.f. to: B(x) = exp( 1/B(x)^3 * Integral B(x)^3 dx ) where B(y) = Bessel polynomial y_n(-3) (cf. A065923).
Note that G(x) = exp(1/G(x)^3 * Integral G(x)^7 dx) has negative coefficients.
CONJECTURE:
Given G(x,n,k) = G such that G = exp( 1/G^n * Integral G^k dx ) then G(x,n,k) consists solely of positive coefficients when k >= A047399(n) where A047399 lists numbers that are congruent to {0,3,6} mod 8.

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 213*x^4/4! + 3096*x^5/5! +...
Related expansions:
log(A(x)) = x + 2*x^2/2! + 17*x^3/3! + 120*x^4/4! + 1905*x^5/5! + 23640*x^6/6! +...
Integral A(x)^8 dx = x + 8*x^2/2! + 80*x^3/3! + 1032*x^4/4! + 16320*x^5/5! +...
1/A(x)^3 = 1 - 3*x + 3*x^2/2! - 24*x^3/3! + 117*x^4/4! - 2088*x^5/5! +...

Paul D. Hanna, Table of n, a(n) for n = 0..300

Cf. A232690, A232691.

seq(n! * coeff(series((3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5), x, n+1), x, n), n=0..20) # Vaclav Kotesovec, Jan 05 2014

m = 20; A[] = 1; Do[A[x] = Exp[1/A[x]^3 Integrate[A[x]^8 + O[x]^m, x]] + O[x]^m // Normal, {m}]; CoefficientList[A[x], x] Range[0, m-1]! (* Jean-François Alcover, Nov 03 2019 *)

{a(n)=local(A=1+x);for(i=1,n,A=exp(1/A^3*intformal(A^8+x*O(x^n))));n!*polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

E.g.f.: (3*LambertW(-1, (25*x-8)/3*exp(-8/3))/(25*x-8))^(1/5). - Vaclav Kotesovec, Jan 05 2014

cp's OEIS Frontend

A232690 E.g.f. satisfies: A(x) = exp( 1/A(x) * Integral A(x)^3 dx ).

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