Search: a266329 -id:a266329
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A266328
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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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+10
4
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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
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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).
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PROG
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(PARI) {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), ", "))
(PARI) {a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) - x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) Vec( serlaplace( exp( serreverse( intformal( 1/(exp(x +x*O(x^25)) - x)))))) \\ Joerg Arndt, Dec 26 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A266490
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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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+10
2
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1, 1, 4, 20, 126, 972, 8876, 93580, 1119328, 14986944, 222184136, 3614288272, 64022264176, 1226914925840, 25295189791296, 558317369479616, 13136590271813856, 328243850207690432, 8680766764223956416, 242245419192494844096, 7113910552105144027136, 219304957649505551899136, 7081169542830272102170752, 238996807468258679150596352
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OFFSET
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0,3
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COMMENTS
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Compare to: G(x) = exp( Integral G(x) dx ) when G(x) = 1/(1-x).
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LINKS
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FORMULA
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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
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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PROG
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(PARI) {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), ", "))
(PARI) {a(n) = n! * polcoeff( exp( serreverse( intformal( 1/(exp(x +x*O(x^n)) + 2*x) ) )), n)}
for(n=0, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A289739
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Expansion of solution to dy/dx = y + exp(y).
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+10
2
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0, 1, 2, 5, 17, 79, 474, 3468, 29799, 293528, 3258373, 40234231, 546921835, 8115147998, 130503876054, 2260929219675, 41979302557200, 831593152814251, 17506400133530765, 390278100156698627, 9185223726173708408, 227578002295869672508, 5921091852493279814589
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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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
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EXAMPLE
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E.g.f. = x + 2*x^2/2! + 5*x^3/3! + 17*x^4/4! + ...
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MAPLE
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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
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MATHEMATICA
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a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ InverseSeries[ Series[Integrate[ 1 / (x + Exp[x]), x], {x, 0, n}]], {x, 0, n}]];
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PROG
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(PARI) {a(n) = if( n<0, 0, my(A = O(x)); for(k=1, n, A = intformal(A + exp(A))); n! * polcoeff(A, n))};
(PARI) {a(n) = if( n<0, 0, n! * polcoeff( serreverse( intformal( 1 / (exp(x + x * O(x^n)) + x))), n))};
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A268170
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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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+10
1
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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
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OFFSET
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0,3
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COMMENTS
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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).
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LINKS
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EXAMPLE
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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).
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PROG
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(PARI) {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), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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