A111670 Array T(n,k) read by antidiagonals: the k-th column contains the first column of the k-th power of A039755.

1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 15, 24, 1, 1, 5, 28, 105, 116, 1, 1, 6, 45, 280, 929, 648, 1, 1, 7, 66, 585, 3600, 9851, 4088, 1, 1, 8, 91, 1056, 9865, 56240, 121071, 28640, 1

Offset

1,5

Comments

First few columns of the array generated from A039775 are: 1, 1, 1, 1, 1, 1, ...; 1, 2, 6, 24, 116, 648, ...; 1, 3, 15, 105, 929, ...; 1, 4, 28, 280, 3600, ...; 1, 5, 45, 585, 9865, ...;

Links

Table of n, a(n) for n=1..45.

Formula

Let A039755 (an analog of Stirling numbers of the second kind) be an infinite lower triangular matrix M; then the vector M^k * [1, 0, 0, 0, ...] (first column of the k-th power) is the k-th column of this array.

Example

 1     1       1          1          1          1          1          1
 1     2       3          4          5          6          7          8
 1     6      15         28         45         66         91        120
 1    24     105        280        585       1056       1729       2640
 1   116     929       3600       9865      22036      43001      76224
 1   648    9851      56240     203565     565096    1318023    2717856
 1  4088  121071    1029920    4953205   17148936   47920803  115146816
 1 28640 1685585   21569600  138529105  600001696 2012844225 5644055040

Maple

A111670 := proc(n, k)
    local A, i, j ;
    A := Matrix(n, n) ;
    for i from 1 to n do
    for j from 1 to n do
        A[i, j] := A039755(i-1, j-1) ;
    end do:
    end do:
    LinearAlgebra[MatrixPower](A, k) ;
    %[n, 1] ;
end proc:
for d from 2 to 12 do
    for n from  1 to d-1 do
        printf("%d, ", A111670(n, d-n)) ;
    end do:
end do: # R. J. Mathar, Jan 27 2023

Mathematica

nmax = 10;
A[n_, k_] := Sum[(-1)^(k-j)*(2j+1)^n*Binomial[k, j], {j, 0, k}]/(2^k*k!);
A039755 = Array[A, {nmax, nmax}, {0, 0}];
T = Table[MatrixPower[A039755, n][[All, 1]], {n, 1, nmax}] // Transpose;
Table[T[[n-k+1, k]], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Apr 02 2024 *)

Crossrefs

Cf. A039755, A007405 (column 2), A000384 (row 2), A011199 (row 3).

Sequence in context: A287024 A107702 A174480 * A123353 A156540 A156582

Adjacent sequences: A111667 A111668 A111669 * A111671 A111672 A111673

Keyword

nonn,tabl

Author

Gary W. Adamson, Aug 14 2005

Extensions

Definition simplified by R. J. Mathar, Jan 27 2023

Status

approved