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Decimal expansion of the number x other than -2 defined by x*exp(x) = -2/e^2.
+10
37
4, 0, 6, 3, 7, 5, 7, 3, 9, 9, 5, 9, 9, 5, 9, 9, 0, 7, 6, 7, 6, 9, 5, 8, 1, 2, 4, 1, 2, 4, 8, 3, 9, 7, 5, 8, 2, 1, 0, 9, 9, 7, 5, 7, 5, 1, 8, 1, 1, 4, 0, 6, 3, 5, 0, 0, 0, 4, 9, 5, 4, 8, 8, 3, 0, 3, 9, 1, 5, 0, 1, 5, 1, 8, 3, 8, 1, 2, 0, 4, 9, 7, 6, 7, 2, 5, 0, 0, 7, 2, 3, 3, 8, 1, 5, 5, 9, 2, 8, 5, 8, 2, 9, 3, 8
FORMULA
Equals LambertW(-2*exp(-2)).
EXAMPLE
-0.4063757399599599076769581241248397582109975751811406350004954883....
MATHEMATICA
RealDigits[N[ProductLog[-2/E^2], 105]][[1]] (* corrected by Vaclav Kotesovec, Feb 21 2014 *)
PROG
(PARI) solve(x=-1, x=0, x*exp(x) + 2*exp(-2)) \\ G. C. Greubel, Nov 15 2017
CROSSREFS
Cf. A106533, A187655, A187657, A217899, A217900, A226750, A243227, A245109, A247238, A258467, A337458.
G.f.: Sum_{n>=0} n^(2*n) * x^n / (1 + n^2*x)^n.
+10
2
1, 1, 15, 602, 46620, 5921520, 1118557440, 294293759760, 102896614941120, 46150861752777600, 25832386565857872000, 17651395149921751680000, 14460364581345685626624000, 13990151265412450143375360000, 15782226575197809064309171200000, 20533602558350213132577801792768000
COMMENTS
Compare to: Sum_{n>=0} n^n * x^n / (1 + n*x)^n = 1 + Sum_{n>=1} (n+1)!/2 * x^n.
FORMULA
a(n) = Sum_{k=0..n-1} (-1)^(n-k-1) * binomial(n-1,k) * (k+1)^(2*n) for n>0 with a(0)=1.
a(n) = (n-1)! * Stirling2(2*n+1, n) for n>0 with a(0)=1.
a(n) = (2*n+1)!/n * [x^(2*n+1)] (exp(x) - 1)^n for n>0 with a(0)=1.
a(n) ~ 2^(2*n+1) * n^(2*n) / (sqrt(1-c) * exp(2*n) * c^n * (2-c)^(n+1)), where c = -LambertW(-2*exp(-2)) = 0.4063757399599599... (see A226775). - Vaclav Kotesovec, Nov 05 2014
EXAMPLE
G.f.: A(x) = 1 + x + 15*x^2 + 602*x^3 + 46620*x^4 + 5921520*x^5 +...
where
A(x) = 1 + x/(1+x) + 4^2*x^2/(1+4*x)^2 + 9^3*x^3/(1+9*x)^3 + 16^4*x^4/(1+16*x)^4 + 25^5*x^5/(1+25*x)^5 + 36^6*x^6/(1+36*x)^6 + 49^7*x^7/(1+49*x)^7 +...
MATHEMATICA
Flatten[{1, Table[(n-1)! * StirlingS2[2*n+1, n], {n, 1, 20}]}] (* Vaclav Kotesovec, Nov 05 2014 *)
PROG
(PARI) {a(n)=polcoeff( sum(m=0, n, m^(2*m)*x^m/(1+m^2*x +x*O(x^n))^m), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n)=if(n==0, 1, sum(k=0, n-1, (-1)^(n-k-1) * binomial(n-1, k) * (k+1)^(2*n)))}
for(n=0, 20, print1(a(n), ", "))
(PARI) {Stirling2(n, k)=n!*polcoeff(((exp(x+x*O(x^n))-1)^k)/k!, n)}
{a(n) = if(n==0, 1, (n-1)! * Stirling2(2*n+1, n) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = if(n==0, 1, (2*n+1)!/n * polcoeff(((exp(x + x*O(x^(2*n+1))) - 1)^n), 2*n+1))}
for(n=0, 20, print1(a(n), ", "))
a(n) = Sum_{k=0..n} (-1)^k * binomial(n, k) * (n - k)^(2*n + 1).
+10
1
0, 1, 30, 1806, 186480, 29607600, 6711344640, 2060056318320, 823172919528960, 415357755774998400, 258323865658578720000, 194165346649139268480000, 173524374976148227519488000, 181871966450361851863879680000, 220951172052769326900328396800000
MAPLE
a := n -> n! * Stirling2(2*n + 1, n):
seq(a(n), n = 0..14);
MATHEMATICA
A367392[n_]:=n!StirlingS2[2n+1, n];
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