A274712 -id:A274712 - cp's OEIS frontend

A226750 Decimal expansion of the number x other than -3 defined by x*e^x = -3/e^3.

1, 7, 8, 5, 6, 0, 6, 2, 7, 8, 7, 7, 9, 2, 1, 1, 0, 6, 5, 9, 6, 8, 0, 8, 6, 6, 9, 7, 0, 5, 5, 1, 4, 8, 0, 4, 6, 5, 4, 1, 1, 8, 2, 5, 5, 9, 2, 6, 8, 8, 5, 9, 0, 7, 7, 2, 0, 1, 4, 2, 3, 0, 6, 0, 5, 8, 7, 8, 5, 6, 9, 5, 4, 9, 4, 4, 8, 2, 6, 0, 8, 7, 5, 4, 3, 1, 3, 4, 3, 4, 6, 5, 2, 1, 6, 0, 3, 5, 5, 6, 1, 0, 4, 0, 0

Offset: 0

José María Grau Ribas, Jun 16 2013

			-0.178560627877921106596808669705514804654118255926885907720142306058785...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A106533, A217913, A221214, A274712, A274713.

RealDigits[N[ProductLog[-3/e^3], 105]][[1]]

solve(x=0,1,3/exp(3)-x*exp(-x)) \\ Charles R Greathouse IV, Nov 19 2013

Term a(104) corrected by G. C. Greubel, Nov 16 2017

A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

Original entry on oeis.org

1, 15, 966, 145750, 40075035, 17505749898, 11143554045652, 9741955019900400, 11201516780955125625, 16392038075086211019625, 29749840488672593296243236, 65580126734167548918100615020, 172597131674172062132363512309613, 534584200037719212882636004559739000, 1924887533450780657560944228447179522880, 7973126100358260458973226689851075932667520, 37645241791600906804871080818625037726247519045

Offset: 1

Paul D. Hanna, Jul 03 2016

nonn

a(n) is divisible by the triangular numbers: a(n) / (n*(n+1)/2) = A274712(n).

			O.g.f.: A(x) = x + 15*x^2 + 966*x^3 + 145750*x^4 + 40075035*x^5 + 17505749898*x^6 + 11143554045652*x^7 + 9741955019900400*x^8 +...
where
A(x) = exp(-x)*x + 2^5*exp(-2^3*x)*x^2/2! + 3^8*exp(-3^3*x)*x^3/3! + 4^11*exp(-4^3*x)*x^4/4! + 5^14*exp(-5^3*x)*x^5/5! + 6^17*exp(-6^3*x)*x^6/6! + 7^20*exp(-7^3*x)*x^7/7! + 8^23*exp(-8^3*x)*x^8/8! +...+ n^(3*n-1)*exp(-n^3*x)*x^n/n! +...
simplifies to an integer series.

Paul D. Hanna, Table of n, a(n) for n = 1..100

Cf. A129506, A217913, A274712, A008277.

Table[StirlingS2[3*n - 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Jul 06 2016 *)

{a(n) = abs( stirling(3*n-1, n, 2) )}
for(n=1, 20, print1(a(n), ", "))

{a(n) = 1/n! * sum(k=0, n, (-1)^(n-k) * binomial(n, k) * k^(3*n-1))}
for(n=1, 20, print1(a(n), ", "))

{a(n) = polcoeff( 1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

{a(n) = polcoeff( sum(m=1, n, m^(3*m-1) * x^m * exp(-m^3*x +x*O(x^n))/m!), n)}
for(n=1, 20, print1(a(n), ", "))

O.g.f.: Sum_{n>=1} n^(3*n-1) * exp(-n^3*x) * x^n / n!, an integer series.
a(n) = A008277(3*n-1,n) for n>=1, where A008277 are the Stirling numbers of the second kind.
a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(3*n-1).
a(n) = [x^(2*n-1)] 1 / Product_{k=1..n} (1 - k*x).
a(n) ~ 3^(3*n-1) * n^(2*n-3/2) / (exp(2*n) * c^n * (3-c)^(2*n-1) * sqrt(2*Pi*(1-c))), where c = -LambertW(-3*exp(-3)) = 0.1785606278779211065968... = -A226750. - Vaclav Kotesovec, Jul 06 2016

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A226750 Decimal expansion of the number x other than -3 defined by x*e^x = -3/e^3.

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A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets.

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