Search: a259062 -id:a259062
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A259063
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E.g.f.: Series_Reversion( 3*x - 2*x*exp(x) ).
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+10
6
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1, 4, 54, 1208, 37810, 1521252, 74800558, 4346473840, 291409650378, 22142153337500, 1880332153123270, 176486211108436968, 18142303135426278562, 2027140583610836224468, 244622970048028087152990, 31706140285613089502561504, 4392907300768938557656691194, 647905974466168686991684285836
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OFFSET
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1,2
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LINKS
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FORMULA
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O.g.f.: x * Sum_{n>=0} 2^n / (3 - n*x)^(n+1).
E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^n / n!.
E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (exp(x)-1)^n * x^(n-1) / n! ).
a(n) ~ (c/(3*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(3*exp(1)/2). - Vaclav Kotesovec, Jun 19 2015
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EXAMPLE
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E.g.f.: A(x) = x + 4*x^2/2! + 54*x^3/3! + 1208*x^4/4! + 37810*x^5/5! + ...
where A(3*x - 2*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 4*x^2 + 54*x^3 + 1208*x^4 + 37810*x^5 + 1521252*x^6 + ...
where F(x)/x = 1/3 + 2/(3-x)^2 + 2^2/(3-2*x)^3 + 2^3/(3-3*x)^4 + 2^4/(3-4*x)^5 +...
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MATHEMATICA
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Rest[CoefficientList[InverseSeries[Series[3*x - 2*x*E^x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)
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PROG
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(PARI) {a(n) = local(A=x); A = serreverse(3*x - 2*x*exp(x +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = x + sum(m=1, n, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = x*exp(sum(m=1, n, Dx(m-1, 2^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, 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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A258872
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E.g.f.: (1/x) * Series_Reversion( -x + 2*x*exp(-x) ).
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+10
1
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1, 2, 14, 182, 3526, 91422, 2978910, 117081974, 5393393078, 285072735950, 17009730803086, 1131081110962662, 82949497319012070, 6651426091458349502, 578967663130916841662, 54369741954121640179286, 5479228772620109128533526, 589841997033535953174559662
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,2
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LINKS
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FORMULA
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E.g.f.: 1 + (1/x)*Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^n / n!.
E.g.f.: exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 2^n * (1 - exp(-x))^n * x^(n-1) / n! ).
a(n) ~ (1-c) * n^(n-1) / (sqrt(1+c) * (c + 1/c - 2)^(n+1) * exp(n)), where c = LambertW(exp(1)/2) = 0.685076942154593946... . - Vaclav Kotesovec, Jun 19 2015
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 14*x^2/2! + 182*x^3/3! + 3526*x^4/4! + 91422*x^5/5! +...
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MATHEMATICA
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CoefficientList[1/x*InverseSeries[Series[-x + 2*x*E^(-x), {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jun 19 2015 *)
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PROG
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(PARI) {a(n) = local(A=x); A = (1/x)*serreverse(-x + 2*x*exp(-x +x^2*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A = 1 + (1/x)*sum(m=1, n, Dx(m-1, 2^m*(1-exp(-x+x^2*O(x^n)))^m*x^m/m!)); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A = exp(sum(m=1, n+1, Dx(m-1, 2^m*(1-exp(-x+x*O(x^n)))^m*x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=0, 25, 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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