

A111933


Triangle read by rows, generated from Stirling cycle numbers.


1



1, 1, 1, 1, 2, 2, 1, 3, 7, 6, 1, 4, 15, 35, 24, 1, 5, 26, 105, 228, 120, 1, 6, 40, 234, 947, 1834, 5040, 1, 7, 57, 440, 2696, 10472, 17582, 40320, 1, 8, 77, 741, 6170, 37919, 137337, 195866, 362880, 1, 9, 100, 1155, 12244, 105315, 630521, 2085605
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OFFSET

1,5


COMMENTS

Rows of the array are generalized coefficients of iterated exponentials: row 1 = (factorials 1, 1, 2, 6...); row 2 = A003713; row 3 = A000268; row 4 = A000310; row 5 = A000359; row 6 = A000406; row 7 = A001765 and so on. Terms in the array, first few rows are: 1, 1, 2, 6, 24, 120,... 1, 2, 7, 35, 228, 1834,... 1, 3, 15, 105, 947, 10472,... 1, 4, 26, 234, 2697, 37919,... 1, 5, 40, 440, 6170, 105315,... 1, 6, 57, 741, 12244, 245755,... ... Column 3 of the array = 2nd pentagonal numbers (2, 7, 15, 26...); column 4 of the array = A094952. First few rows of the triangle are: 1; 1, 1; 1, 2, 2; 1, 3, 7, 6; 1, 4, 15, 35, 24; 1, 5, 26, 105, 228, 120; 1, 6, 40, 234, 947, 1834, 5040; ...


LINKS

Table of n, a(n) for n=1..53.


FORMULA

Let M = the infinite lower triangular matrix of Stirling cycle numbers (A008275). Perform M^n * [1, 0, 0, 0...] forming an array. Antidiagonals of the array become the rows of A111933.


EXAMPLE

Row 5 of the triangle = 1, 4, 15, 35, 24; generated from M^n * [1,0,0,0...] (n = 1 through 5); then take antidiagonals.


CROSSREFS

Cf. A008275, A005449, A094952, A003713, A000268, A000310, A000359, A000406, A011765.
Sequence in context: A330141 A007441 A289192 * A144304 A122941 A297622
Adjacent sequences: A111930 A111931 A111932 * A111934 A111935 A111936


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, Aug 21 2005


STATUS

approved



