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

%I A111670 #24 Jul 23 2024 03:47:07
%S A111670 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,
%T A111670 1,1,7,66,585,3600,9851,4088,1,1,8,91,1056,9865,56240,121071,28640,1
%N A111670 Array T(n,k) read by antidiagonals: the k-th column contains the first column of the k-th power of A039755.
%F A111670 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.
%e A111670  1     1       1          1          1          1          1          1
%e A111670  1     2       3          4          5          6          7          8
%e A111670  1     6      15         28         45         66         91        120
%e A111670  1    24     105        280        585       1056       1729       2640
%e A111670  1   116     929       3600       9865      22036      43001      76224
%e A111670  1   648    9851      56240     203565     565096    1318023    2717856
%e A111670  1  4088  121071    1029920    4953205   17148936   47920803  115146816
%e A111670  1 28640 1685585   21569600  138529105  600001696 2012844225 5644055040
%p A111670 A111670 := proc(n,k)
%p A111670     local A,i,j ;
%p A111670     A := Matrix(n,n) ;
%p A111670     for i from 1 to n do
%p A111670     for j from 1 to n do
%p A111670         A[i,j] := A039755(i-1,j-1) ;
%p A111670     end do:
%p A111670     end do:
%p A111670     LinearAlgebra[MatrixPower](A,k) ;
%p A111670     %[n,1] ;
%p A111670 end proc:
%p A111670 for d from 2 to 12 do
%p A111670     for n from  1 to d-1 do
%p A111670         printf("%d,",A111670(n,d-n)) ;
%p A111670     end do:
%p A111670 end do: # _R. J. Mathar_, Jan 27 2023
%t A111670 nmax = 10;
%t A111670 A[n_, k_] := Sum[(-1)^(k-j)*(2j+1)^n*Binomial[k, j], {j, 0, k}]/(2^k*k!);
%t A111670 A039755 = Array[A, {nmax, nmax}, {0, 0}];
%t A111670 T = Table[MatrixPower[A039755, n][[All, 1]], {n, 1, nmax}] // Transpose;
%t A111670 Table[T[[n-k+1, k]], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Apr 02 2024 *)
%Y A111670 Cf. A039755, A007405 (column 2), A000384 (row 2), A011199 (row 3).
%K A111670 nonn,tabl
%O A111670 1,5
%A A111670 _Gary W. Adamson_, Aug 14 2005
%E A111670 Definition simplified by _R. J. Mathar_, Jan 27 2023