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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. 12
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
LINKS
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: A213191 A352449 A079520 * A208981 A357892 A261158
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 10 2013
STATUS
approved

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Last modified March 2 04:27 EST 2024. Contains 370460 sequences. (Running on oeis4.)