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A111670
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Array T(n,k) read by antidiagonals: the k-th column contains the first column of the k-th power of A039755.
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1
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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
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OFFSET
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1,5
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COMMENTS
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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, ...;
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LINKS
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Table of n, a(n) for n=1..45.
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FORMULA
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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.
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EXAMPLE
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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
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MAPLE
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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
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CROSSREFS
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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
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KEYWORD
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nonn,tabl,changed
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AUTHOR
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Gary W. Adamson, Aug 14 2005
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EXTENSIONS
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Definition simplified by R. J. Mathar, Jan 27 2023
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STATUS
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approved
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