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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

[] is an Iverson bracket.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

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-Fran├žois Alcover, Feb 09 2021 *)

CROSSREFS

Columns k=0-2 give: A000012, A007526, 2*A134432(n-1).

Main diagonal gives A000142.

Row sums give A308876.

Cf. A000522, A073474, A093964, A143409, A196347, A271705, A327606.

Sequence in context: A171650 A225476 A143777 * A236830 A269736 A264535

Adjacent sequences:  A326656 A326657 A326658 * A326660 A326661 A326662

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 12 2019

STATUS

approved

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Last modified June 16 09:57 EDT 2021. Contains 345056 sequences. (Running on oeis4.)