login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A274713 Number of partitions of a {3*n-1}-set into n nonempty subsets. 3
1, 15, 966, 145750, 40075035, 17505749898, 11143554045652, 9741955019900400, 11201516780955125625, 16392038075086211019625, 29749840488672593296243236, 65580126734167548918100615020, 172597131674172062132363512309613, 534584200037719212882636004559739000, 1924887533450780657560944228447179522880, 7973126100358260458973226689851075932667520, 37645241791600906804871080818625037726247519045 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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

LINKS

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

(PARI) {a(n) = abs( stirling(3*n-1, n, 2) )}

for(n=1, 20, print1(a(n), ", "))

(PARI) {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), ", "))

(PARI) {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), ", "))

(PARI) {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), ", "))

CROSSREFS

Cf. A129506, A217913, A274712, A008277.

Sequence in context: A103639 A055413 A067408 * A229840 A102102 A196569

Adjacent sequences:  A274710 A274711 A274712 * A274714 A274715 A274716

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 03 2016

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 8 14:38 EST 2019. Contains 329865 sequences. (Running on oeis4.)