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A232473 3-Fubini numbers. 5
6, 42, 342, 3210, 34326, 413322, 5544342, 82077450, 1330064406, 23428165002, 445828910742, 9116951060490, 199412878763286, 4646087794988682, 114884369365147542, 3005053671533400330, 82905724863616146966, 2406054103612912660362, 73277364784409578094742, 2336825320400166931304970 (list; graph; refs; listen; history; text; internal format)
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.

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-Fran├žois 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

AUTHOR

N. J. A. Sloane, Nov 27 2013

STATUS

approved

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Last modified July 25 03:36 EDT 2021. Contains 346282 sequences. (Running on oeis4.)