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.

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

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-François 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