OFFSET
4,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 4..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+4) = Sum_{k = 0..n} (k+4)!/k!*( Sum{i = 0..k} (-1)^(k-i)*binomial(k,i)*(i+4)^n ).
a(n+4) = Sum_{k = 0..n} 4^(n-k)*binomial(n,k)*( Sum_{i = 0..k} Stirling2(k,i)*(i+4)! ).
E.g.f. with offset 0: 24*exp(4*z)/(2 - exp(z))^5 = 24 + 216*z + 2184*z^2/2! + 24696*z^3/3! + .... (End)
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Dec 17 2020
MAPLE
T := (n, k, r) -> (1/(k-r)!)*add ((-1)^(k+i+r)*binomial(k-r, i)*(i+r)^(n-r), i = 0..k-r):
B := (n, r) -> add(T(n, k, r), k=r..n);
SB := r -> [seq(B(n, r), n=r..30)];
SB(2);
F := (n, r) -> add((k)!*T(n, k, r), k=r..n);
SF := r -> [seq(F(n, r), n=r..30)];
SF(4);
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, 4], {n, 4, 22}] (* Jean-François Alcover, Mar 30 2016 *)
PROG
(Magma) r:=4; 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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Nov 27 2013
STATUS
approved