A229003 -id:A229003

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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.

+10
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

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.

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 10 2013

STATUS

approved

A229091

a(n) = ((-1)^n*(2^n-1) + Sum_{k>=1} (k^n*(k^2+k-1)/(k+2)!))/exp(1).

+10
1

0, 2, 0, 14, 20, 152, 532, 2914, 14604, 83342, 494164, 3127016, 20810088, 145645866, 1067655656, 8177942670, 65292914084, 542226906224, 4674687594572, 41766307038106, 386112935883604, 3687989974641678, 36347655981682676, 369185211517110928, 3860146249155022160

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

OFFSET

1,2

COMMENTS

Sequence is related to asymptotic of A229001.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..300

FORMULA

a(n) = Bell(n) - Bell(n+1) + Sum_{j=0..n} ((-1)^j*(2^j*((2*n-j+1)/(j+1))-1) * Bell(n-j) * C(n,j)).

EXAMPLE

Sequence A228997 (column k=7 of A229001) is asymptotic to n!*(532*exp(1)+127)*n, therefore a(7) = 532.

MATHEMATICA

Table[Simplify[((-1)^n*(2^n-1) + Sum[k^n*(k^2+k-1)/(k+2)!, {k, 1, Infinity}])/E], {n, 1, 20}] (* from definition *)

Table[BellB[n] - BellB[n+1] + Sum[(-1)^j*(2^j*((2*n-j+1)/(j+1))-1) * BellB[n-j]*Binomial[n, j], {j, 0, n}], {n, 1, 20}] (* faster *)

CROSSREFS

Cf. A229001, A228959, A229003, A228994, A228995, A228996, A228997, A228998, A228999, A229000.

KEYWORD

nonn

AUTHOR

Vaclav Kotesovec, Sep 13 2013

STATUS

approved

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