A266329 -id:A266329 - cp's OEIS frontend

A266328 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(-x) * exp( Integral A(x) dx ), where the constant of integration is zero.

1, 1, 1, 2, 6, 21, 92, 469, 2731, 17985, 131528, 1059616, 9319363, 88833422, 912393381, 10043727089, 117969438513, 1472593659884, 19467505081458, 271704942613323, 3992343851680466, 61603531051030691, 995949139457447931, 16835191741257445589, 296976010796327785530, 5457427389713208932740, 104308245862443706265341, 2070461793105333579698992, 42622090166454492404075635

Offset: 0

Paul D. Hanna, Jan 24 2016

nonn

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).
What is lim_{n->oo} (a(n)/n!)^(1/n)?  Example: (a(500)/500!)^(1/500) = 0.7353325805...
Limit_{n->oo} (a(n)/n!)^(1/n) = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.73578196429164719984313538... - Vaclav Kotesovec, Aug 21 2017

			E.g.f.: A(x) = 1 + x + x^2/2! + 2*x^3/3! + 6*x^4/4! + 21*x^5/5! + 92*x^6/6! + 469*x^7/7! + 2731*x^8/8! + 17985*x^9/9! + 131528*x^10/10! + ...
such that log(A(x)) = Integral B(x) dx
where
B(x) = 1 + x^2/2! + x^3/3! + 5*x^4/4! + 16*x^5/5! + 76*x^6/6! + 393*x^7/7! + 2338*x^8/8! + 15647*x^9/9! + 115881*x^10/10! + ...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) + 1,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) - log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx - x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) - x) dx:
Integral 1/(exp(x) - x) dx  =  x - x^3/3! - x^4/4! + 5*x^5/5! + 19*x^6/6! - 41*x^7/7! - 519*x^8/8! - 183*x^9/9! + 19223*x^10/10! + ... + A089148(n-1)*x^n/n! + ...
so that A( Integral 1/(exp(x) - x) dx ) = exp(x).

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

Cf. A006789, A089148, A266329.

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

{a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) - x) ) )), n)}
for(n=0,30,print1(a(n),", "))

Vec( serlaplace( exp( serreverse( intformal( 1/(exp(x +x*O(x^25)) - x)))))) \\ Joerg Arndt, Dec 26 2023

E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) - log(A(x)) dx ).
(2) A(x) = log(A(x)) + A'(x)/A(x).
(3) log(A(x)) = exp(-x) * Integral exp(x)*A(x) dx.
(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) - x) dx ) ).
a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..oo} 1/(exp(x) - x) dx = 0.7357819642916471998431353808137704665788888148929882090175... - Vaclav Kotesovec, Aug 21 2017
Conjecture: a(n) = R(n-1, 0) for n > 0 with a(0) = 1 where R(n, q) = R(n-1, q+1) + Sum_{j=0..q-1} binomial(q+1, j)*R(n-1, j) for n > 0, q >= 0 with R(0, q) = 1 for q >= 0. - Mikhail Kurkov, Dec 26 2023

A266490 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(2x) exp( Integral A(x) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, 1, 4, 20, 126, 972, 8876, 93580, 1119328, 14986944, 222184136, 3614288272, 64022264176, 1226914925840, 25295189791296, 558317369479616, 13136590271813856, 328243850207690432, 8680766764223956416, 242245419192494844096, 7113910552105144027136, 219304957649505551899136, 7081169542830272102170752, 238996807468258679150596352

Offset: 0

Paul D. Hanna, Jan 27 2016

nonn

Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 20*x^3/3! + 126*x^4/4! + 972*x^5/5! + 8876*x^6/6! + 93580*x^7/7! + 1119328*x^8/8! + 14986944*x^9/9! + 222184136*x^10/10! +...
such that log(A(x)) = Integral B(x) dx
where B(x) = 1 + 3*x + 10*x^2/2! + 40*x^3/3! + 206*x^4/4! + 1384*x^5/5! + 11644*x^6/6! + 116868*x^7/7! + 1353064*x^8/8! + 17693072*x^9/9! + 257570280*x^10/10! +...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) - 2,
(2) B(x) = A'(x)/A(x),
(3) B(x) = A(x) + 2*log(A(x)),
(4) log(A(x)) = Integral B(x) dx,
(5) log(B(x)) = Integral A(x) dx + 2*x.
The Series Reversion of log(A(x)) equals Integral 1/(exp(x) + 2*x) dx:
Integral 1/(exp(x) + 2*x) dx  =  x - 3*x^2/2! + 17*x^3/3! - 145*x^4/4! + 1649*x^5/5! - 23441*x^6/6! + 399865*x^7/7! - 7957881*x^8/8! + 180997857*x^9/9! - 4631289697*x^10/10! +...
so that A( Integral 1/(exp(x) + 2*x) dx ) = exp(x).

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A266328, A266329.

a[ n_] := a[n] = If[ n < 1, Boole[n == 0], Sum[ Binomial[n - 1, k - 1] a[n - k] Sum[ 2^(j - 1) a[k - j], {j, k}], {k, n}]]; (* Michael Somos, Aug 08 2017 *)

{a(n) = my(A=1+x, B=1+x); for(i=0, n, A = exp( intformal( B + x*O(x^n) ) ); B = exp( intformal( 2 + A ) ) ); n!*polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))

{a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) + 2*x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))

E.g.f. A(x) satisfies:
(1) A(x) = exp( Integral A(x) + 2*log(A(x)) dx ).
(2) A(x) = A'(x)/A(x) - 2*log(A(x)).
(3) log(A(x)) = exp(2*x) * Integral exp(-2*x)*A(x) dx.
(4) A(x) = exp( Series_Reversion( Integral 1/(exp(x) + 2*x) dx ) ).
a(n) ~ c^(n+1) * n!, where c = 1/Integral_{x=0..infinity} 1/(2*x + exp(x)) dx = 1.4650202775490107369040248583790383461628786237838809798971... - Vaclav Kotesovec, Aug 21 2017

A289739 Expansion of solution to dy/dx = y + exp(y).

Original entry on oeis.org

0, 1, 2, 5, 17, 79, 474, 3468, 29799, 293528, 3258373, 40234231, 546921835, 8115147998, 130503876054, 2260929219675, 41979302557200, 831593152814251, 17506400133530765, 390278100156698627, 9185223726173708408, 227578002295869672508, 5921091852493279814589

Offset: 0

Michael Somos, Aug 09 2017

nonn

See A266539 for more details.

			E.g.f. = x + 2*x^2/2! + 5*x^3/3! + 17*x^4/4! + ...

Robert Israel, Table of n, a(n) for n = 0..435

Cf. A266329.

S:= dsolve({diff(y(x),x) = y(x) + exp(y(x)), y(0)=0},y(x),series,order=31):
seq(coeff(rhs(S),x,j)*j!,j=0..30); # Robert Israel, Aug 09 2017

a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ InverseSeries[ Series[Integrate[ 1 / (x + Exp[x]), x], {x, 0, n}]], {x, 0, n}]];

{a(n) = if( n<0, 0, my(A = O(x)); for(k=1, n, A = intformal(A + exp(A))); n! * polcoeff(A, n))};

{a(n) = if( n<0, 0, n! * polcoeff( serreverse( intformal( 1 / (exp(x + x * O(x^n)) + x))), n))};

E.g.f. y(x) = log(A(x)) and y'(x) = B(x) where A(x), B(x) are as in A266539.

a(n) ~ c^n * (n-1)!, where c = 1/Integral_{x=0..infinity} 1/(x + exp(x)) dx = 1.2400861064984976662394901721056528110217273471501174317019052800276... - Vaclav Kotesovec, Aug 21 2017

A268170 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(1+x - exp(x)) * exp( Integral A(x) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, 1, 2, 5, 16, 65, 326, 1947, 13410, 104181, 900214, 8566655, 89055224, 1004141647, 12204369138, 159036267519, 2211764983734, 32696763676521, 511987792322430, 8465194670035767, 147370831072230860, 2694506417687396995, 51622643862824956898, 1034153511794063402519, 21621325640846679627146

Offset: 0

Paul D. Hanna, Jan 27 2016

nonn

Compare to: F(x) = exp( Integral G(x) dx ) such that G(x) = exp(1-exp(x)) * exp( Integral F(x) dx ) holds when F(x) = exp(x).

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 5*x^3/3! + 16*x^4/4! + 65*x^5/5! + 326*x^6/6! + 1947*x^7/7! + 13410*x^8/8! + 104181*x^9/9! + 900214*x^10/10! + 8566655*x^11/11! +...
such that log(A(x)) = Integral B(x) dx
where
B(x) = 1 + x + x^2/2! + 2*x^3/3! + 9*x^4/4! + 46*x^5/5! + 245*x^6/6! + 1474*x^7/7! + 10315*x^8/8! + 82174*x^9/9! + 726591*x^10/10! + 7038632*x^11/11! + 74216949*x^12/12! +...+ A268171(n)*x^n/n! +...
and A(x) and B(x) satisfy:
(1) A(x) = B'(x)/B(x) + exp(x) - 1,
(2) B(x) = A'(x)/A(x),
(3) log(A(x)) = Integral B(x) dx,
(4) log(B(x)) = Integral A(x) dx + 1+x - exp(x).
RELATED SERIES.
log(A(x)) = x + x^2/2! + x^3/3! + 2*x^4/4! + 9*x^5/5! + 46*x^6/6! + 245*x^7/7! + 1474*x^8/8! + 10315*x^9/9! + 82174*x^10/10! + 726591*x^11/11! + 7038632*x^12/12! +...
Let J(x) equal the series reversion of log(A(x)); then
J(x) = x - x^2/2! + 2*x^3/3! - 7*x^4/4! + 31*x^5/5! - 172*x^6/6! + 1155*x^7/7! - 9027*x^8/8! + 80676*x^9/9! - 811727*x^10/10! + 9075333*x^11/11! - 111633356*x^12/12! +...
where A(J(x)) = exp(x).

Cf. A266328, A266329, A266490, A268171.

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

cp's OEIS Frontend

A266328 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(-x) * exp( Integral A(x) dx ), where the constant of integration is zero.

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A266490 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(2*x) * exp( Integral A(x) dx ), where the constant of integration is zero.

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A289739 Expansion of solution to dy/dx = y + exp(y).

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A268170 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(1+x - exp(x)) * exp( Integral A(x) dx ), where the constant of integration is zero.

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A266490 E.g.f. A(x) satisfies: A(x) = exp( Integral B(x) dx ) such that B(x) = exp(2x) exp( Integral A(x) dx ), where the constant of integration is zero.