A229001 Total sum A(n,k) of the k-th powers of lengths of ascending runs in all permutations of [n]; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 3, 0, 1, 4, 12, 0, 1, 6, 18, 60, 0, 1, 10, 32, 96, 360, 0, 1, 18, 66, 186, 600, 2520, 0, 1, 34, 152, 426, 1222, 4320, 20160, 0, 1, 66, 378, 1110, 2964, 9086, 35280, 181440, 0, 1, 130, 992, 3186, 8254, 22818, 75882, 322560, 1814400

Offset

0,6

Links

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

Formula

A(n,k) = Sum_{t=1..n} t^k * A122843(n,t).

For fixed k, A(n,k) ~ n! * n * sum(t>=1, t^k*(t^2+t-1)/(t+2)!) = n! * n * ((Bell(k) - Bell(k+1) + sum(j=0..k, (-1)^j*(2^j*((2*k-j+1)/(j+1))-1) *Bell(k-j)*C(k,j)))*exp(1) - (-1)^k*(2^k-1)), where Bell(k) are Bell numbers A000110. - Vaclav Kotesovec, Sep 12 2013

Example

A(3,2) = 32 = 9+5+5+5+5+3 = 3^2+4*(2^2+1^2)+3*1^2: (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), (3,2,1).
Square array A(n,k) begins:
:    0,    0,    0,     0,     0,      0,      0, ...
:    1,    1,    1,     1,     1,      1,      1, ...
:    3,    4,    6,    10,    18,     34,     66, ...
:   12,   18,   32,    66,   152,    378,    992, ...
:   60,   96,  186,   426,  1110,   3186,   9846, ...
:  360,  600, 1222,  2964,  8254,  25620,  86782, ...
: 2520, 4320, 9086, 22818, 66050, 214410, 765506, ...

Maple

A:= (n, k)-> add(`if`(n=t, 1, n!/(t+1)!*(t*(n-t+1)+1
             -((t+1)*(n-t)+1)/(t+2)))*t^k, t=1..n):
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := Sum[If[n == t, 1, n!/(t + 1)!*(t*(n - t + 1) + 1 - ((t + 1)*(n - t) + 1)/(t + 2))]* t^k, {t, 1, n}]; Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Crossrefs

Columns k=0-10 give: A001710(n+1) for n>0, A001563, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000.

Rows n=0-2 give: A000004, A000012, A052548.

Main diagonal gives: A229002.

Sequence in context: A352449 A375546 A079520 * A208981 A357892 A261158

Adjacent sequences: A228998 A228999 A229000 * A229002 A229003 A229004

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 10 2013

Status

approved