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A326659
T(n,k) = [0<k<=n] * n*(T(n-1,k-1)+T(n-1,k)) + [k=0 and n>=0]; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows.
5
1, 1, 1, 1, 4, 2, 1, 15, 18, 6, 1, 64, 132, 96, 24, 1, 325, 980, 1140, 600, 120, 1, 1956, 7830, 12720, 10440, 4320, 720, 1, 13699, 68502, 143850, 162120, 103320, 35280, 5040, 1, 109600, 657608, 1698816, 2447760, 2123520, 1108800, 322560, 40320
OFFSET
0,5
COMMENTS
[] is an Iverson bracket.
LINKS
Wikipedia, Iverson bracket
FORMULA
E.g.f. of column k: exp(x)*(x/(1-x))^k.
T(n,k) = k! * A271705(n,k).
T(n,k) = n * A073474(n-1,k-1) for n,k >= 1.
T(n,1) = n * A000522(n-1) for n >= 1.
T(n,2) = n * A093964(n-1) for n >= 1.
Sum_{k=1..n} k * T(n,k) = A327606(n).
EXAMPLE
Triangle T(n,k) begins:
1;
1, 1;
1, 4, 2;
1, 15, 18, 6;
1, 64, 132, 96, 24;
1, 325, 980, 1140, 600, 120;
1, 1956, 7830, 12720, 10440, 4320, 720;
1, 13699, 68502, 143850, 162120, 103320, 35280, 5040;
...
MAPLE
T:= proc(n, k) option remember;
`if`(0<k and k<=n, n*(T(n-1, k-1)+T(n-1, k)), 0)+
`if`(k=0 and n>=0, 1, 0)
end:
seq(seq(T(n, k), k=0..n), n=0..10);
MATHEMATICA
T[n_ /; n >= 0, k_ /; k >= 0] := T[n, k] = Boole[0 < k <= n]*n*(T[n-1, k-1] + T[n-1, k]) + Boole[k == 0 && n >= 0];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 09 2021 *)
CROSSREFS
Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).
Main diagonal gives A000142.
Row sums give A308876.
Sequence in context: A225476 A143777 A365566 * A236830 A269736 A264535
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 12 2019
STATUS
approved