login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

3-Fubini numbers.
5

%I #33 Sep 13 2024 16:27:48

%S 6,42,342,3210,34326,413322,5544342,82077450,1330064406,23428165002,

%T 445828910742,9116951060490,199412878763286,4646087794988682,

%U 114884369365147542,3005053671533400330,82905724863616146966,2406054103612912660362,73277364784409578094742,2336825320400166931304970

%N 3-Fubini numbers.

%H Vincenzo Librandi, <a href="/A232473/b232473.txt">Table of n, a(n) for n = 3..200</a>

%H 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 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 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 From _Peter Bala_, Dec 16 2020: (Start)

%F a(n+3) = Sum_{k = 0..n} (k+3)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+3)^n ).

%F a(n+3) = Sum_{k = 0..n} 3^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+3)! ).

%F 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 a(n) ~ n! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Dec 17 2020

%p # r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):

%p 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 # r-Bell numbers (e.g. A000110, A005493, A005494):

%p B := (n,r) -> add(T(n,k,r),k=r..n);

%p SB := r -> [seq(B(n,r),n=r..30)];

%p SB(2);

%p # r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):

%p F := (n,r) -> add((k)!*T(n,k,r),k=r..n);

%p SF := r -> [seq(F(n,r),n=r..30)];

%p SF(3);

%t 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 (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 [r_Fubini(n, r): n in [r..22]]; // _Bruno Berselli_, Mar 30 2016

%Y Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A226738, A232472, A232473, A232474.

%K nonn,easy

%O 3,1

%A _N. J. A. Sloane_, Nov 27 2013