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A335545 A(n,k) is the sum of the k-th powers of the (positive) number of permutations of [n] with j descents (j=0..max(0,n-1)); square array A(n,k), n>=0, k>=0, read by antidiagonals. 4
1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 6, 4, 1, 1, 2, 18, 24, 5, 1, 1, 2, 66, 244, 120, 6, 1, 1, 2, 258, 2664, 5710, 720, 7, 1, 1, 2, 1026, 29284, 322650, 188908, 5040, 8, 1, 1, 2, 4098, 322104, 19888690, 55457604, 8702820, 40320, 9, 1, 1, 2, 16386, 3543124, 1276095330, 16657451236, 17484605040, 524888040, 362880, 10 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

Alois P. Heinz, Antidiagonals n = 0..60, flattened

FORMULA

A(n,k) = Sum_{j=0..max(0,n-1)} A173018(n,j)^k.

EXAMPLE

Square array A(n,k) begins:

  1,   1,      1,        1,           1,             1, ...

  1,   1,      1,        1,           1,             1, ...

  2,   2,      2,        2,           2,             2, ...

  3,   6,     18,       66,         258,          1026, ...

  4,  24,    244,     2664,       29284,        322104, ...

  5, 120,   5710,   322650,    19888690,    1276095330, ...

  6, 720, 188908, 55457604, 16657451236, 5025377832180, ...

  ...

MAPLE

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,

      expand(add(b(u-j, o+j-1, 1)*x^t, j=1..u))+

             add(b(u+j-1, o-j, 1), j=1..o))

    end:

A:= (n, k)-> (p-> add(coeff(p, x, i)^k, i=0..degree(p)))(b(n, 0$2)):

seq(seq(A(n, d-n), n=0..d), d=0..10);

# second Maple program:

A:= (n, k)-> add(combinat[eulerian1](n, j)^k, j=0..max(0, n-1)):

seq(seq(A(n, d-n), n=0..d), d=0..10);

MATHEMATICA

B[n_, k_] := B[n, k] = Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];

A[0, _] = 1; A[n_, k_] := Sum[B[n, j]^k, {j, 0, n-1}];

Table[A[n, d-n], {d, 0, 10}, {n, 0, d}] // Flatten (* Jean-Fran├žois Alcover, Feb 11 2021 *)

CROSSREFS

Columns k=0-2 give: A028310, A000142, A168562.

Rows n=0+1, 2-3 give: A000012, A007395(k+1), A178789(k+1).

Main diagonal gives A335546.

Cf. A008292, A173018, A334622.

Sequence in context: A116855 A173265 A157744 * A334997 A030111 A096921

Adjacent sequences:  A335542 A335543 A335544 * A335546 A335547 A335548

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 12 2020

STATUS

approved

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Last modified September 24 15:28 EDT 2021. Contains 347643 sequences. (Running on oeis4.)