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A232473 3-Fubini numbers.

%I A232473 #33 Sep 13 2024 16:27:48
%S A232473 6,42,342,3210,34326,413322,5544342,82077450,1330064406,23428165002,
%T A232473 445828910742,9116951060490,199412878763286,4646087794988682,
%U A232473 114884369365147542,3005053671533400330,82905724863616146966,2406054103612912660362,73277364784409578094742,2336825320400166931304970
%N A232473 3-Fubini numbers.
%H A232473 Vincenzo Librandi, <a href="/A232473/b232473.txt">Table of n, a(n) for n = 3..200</a>
%H A232473 Andrei Z. Broder, <a href="http://dx.doi.org/10.1016/0012-365X(84)90161-4">The r-Stirling numbers</a>, Discrete Math. 49, 241-259 (1984).
%H A232473 I. Mezo, <a href="http://arxiv.org/abs/1308.1637">Periodicity of the last digits of some combinatorial sequences</a>, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
%H A232473 Benjamin Schreyer, <a href="https://arxiv.org/abs/2409.03799">Rigged Horse Numbers and their Modular Periodicity</a>, arXiv:2409.03799 [math.CO], 2024. See p. 12.
%F A232473 From _Peter Bala_, Dec 16 2020: (Start)
%F A232473 a(n+3) = Sum_{k = 0..n} (k+3)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n ).
%F A232473 a(n+3) = Sum_{k = 0..n} 3^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+3)! ).
%F A232473 E.g.f. with offset 0: 6*exp(3*z)/(2 - exp(z))^4 = 6 + 42*z + 342*z^2/2! + 3210*z^3/3! + .... (End)
%F A232473 a(n) ~ n! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Dec 17 2020
%p A232473 # r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):
%p A232473 T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
%p A232473 # r-Bell numbers (e.g. A000110, A005493, A005494):
%p A232473 B := (n,r) -> add(T(n,k,r),k=r..n);
%p A232473 SB := r -> [seq(B(n,r),n=r..30)];
%p A232473 SB(2);
%p A232473 # r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):
%p A232473 F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
%p A232473 SF := r -> [seq(F(n,r),n=r..30)];
%p A232473 SF(3);
%t A232473 Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Table[Fubini[n, 3], {n, 3, 22}] (* _Jean-François Alcover_, Mar 30 2016 *)
%o A232473 (Magma) r:=3; r_Fubini:=func<n,r | &+[Factorial(k)*&+[(-1)^(k+h+r)*(h+r)^(n-r)/(Factorial(h)*Factorial(k-h-r)): h in [0..k-r]]: k in [r..n]]>;
%o A232473 [r_Fubini(n, r): n in [r..22]]; // _Bruno Berselli_, Mar 30 2016
%Y A232473 Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A226738, A232472, A232473, A232474.
%K A232473 nonn,easy
%O A232473 3,1
%A A232473 _N. J. A. Sloane_, Nov 27 2013