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A232473
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3-Fubini numbers.
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5
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6, 42, 342, 3210, 34326, 413322, 5544342, 82077450, 1330064406, 23428165002, 445828910742, 9116951060490, 199412878763286, 4646087794988682, 114884369365147542, 3005053671533400330, 82905724863616146966, 2406054103612912660362, 73277364784409578094742, 2336825320400166931304970
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OFFSET
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3,1
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LINKS
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FORMULA
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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)
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MAPLE
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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(3);
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MATHEMATICA
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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 *)
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PROG
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(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]]>;
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CROSSREFS
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Cf. A008277, A143494, A143495, A000110, A005493, A005494, A000670, A226738, A232472, A232473, A232474.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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