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

3,1

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.

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A123510 A265871 A132804 * A300693 A222062 A074017

Adjacent sequences: A232470 A232471 A232472 * A232474 A232475 A232476

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, Nov 27 2013

Status

approved