A232473 3-Fubini numbers.

6, 42, 342, 3210, 34326, 413322, 5544342, 82077450, 1330064406, 23428165002, 445828910742, 9116951060490, 199412878763286, 4646087794988682, 114884369365147542, 3005053671533400330, 82905724863616146966, 2406054103612912660362, 73277364784409578094742, 2336825320400166931304970

Offset: 3

Author

N. J. A. Sloane, Nov 27 2013

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 3..200
• Andrei Z. Broder, The r-Stirling numbers, Discrete Math. 49, 241-259 (1984).
• I. Mezo, Periodicity of the last digits of some combinatorial sequences, arXiv preprint arXiv:1308.1637 [math.CO], 2013.
• Benjamin Schreyer, Rigged Horse Numbers and their Modular Periodicity, arXiv:2409.03799 [math.CO], 2024. See p. 12.

Crossrefs

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

Programs

Magma

r:=3; r_Fubini:=func;
[r_Fubini(n, r): n in [r..22]]; // Bruno Berselli, Mar 30 2016

Maple

# r-Stirling numbers of second kind (e.g. A008277, A143494, A143495):
T := (n,k,r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r,i)*(i+r)^(n-r),i = 0..k-r):
# r-Bell numbers (e.g. A000110, A005493, A005494):
B := (n,r) -> add(T(n,k,r),k=r..n);
SB := r -> [seq(B(n,r),n=r..30)];
SB(2);
# r-Fubini numbers (e.g. A000670, A232472, A232473, A232474):
F := (n,r) -> add((k)!*T(n,k,r),k=r..n);
SF := r -> [seq(F(n,r),n=r..30)];
SF(3);

Mathematica

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 *)

Formula

From Peter Bala, Dec 16 2020: (Start)

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

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

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)

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020