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

%I

%S 1,15,966,145750,40075035,17505749898,11143554045652,9741955019900400,

%T 11201516780955125625,16392038075086211019625,

%U 29749840488672593296243236,65580126734167548918100615020,172597131674172062132363512309613,534584200037719212882636004559739000,1924887533450780657560944228447179522880,7973126100358260458973226689851075932667520,37645241791600906804871080818625037726247519045

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

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

%H Paul D. Hanna, <a href="/A274713/b274713.txt">Table of n, a(n) for n = 1..100</a>

%F O.g.f.: Sum_{n>=1} n^(3*n-1) * exp(-n^3*x) * x^n / n!, an integer series.

%F a(n) = A008277(3*n-1,n) for n>=1, where A008277 are the Stirling numbers of the second kind.

%F a(n) = 1/n! * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * k^(3*n-1).

%F a(n) = [x^(2*n-1)] 1 / Product_{k=1..n} (1 - k*x).

%F 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

%e 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 +...

%e where

%e 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! +...

%e simplifies to an integer series.

%t Table[StirlingS2[3*n - 1, n], {n, 1, 20}] (* _Vaclav Kotesovec_, Jul 06 2016 *)

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

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

%o (PARI) {a(n) = 1/n! * sum(k=0, n, (-1)^(n-k) * binomial(n, k) * k^(3*n-1))}

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

%o (PARI) {a(n) = polcoeff( 1/prod(k=1, n, 1-k*x +x*O(x^(2*n))), 2*n-1)}

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

%o (PARI) {a(n) = polcoeff( sum(m=1, n, m^(3*m-1) * x^m * exp(-m^3*x +x*O(x^n))/m!), n)}

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

%Y Cf. A129506, A217913, A274712, A008277.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 03 2016

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Last modified January 21 16:22 EST 2020. Contains 331114 sequences. (Running on oeis4.)