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A119502
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Triangle read by rows, T(n,k) = (n-k)!, for n>=0 and 0<=k<=n.
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4
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1, 1, 1, 2, 1, 1, 6, 2, 1, 1, 24, 6, 2, 1, 1, 120, 24, 6, 2, 1, 1, 720, 120, 24, 6, 2, 1, 1, 5040, 720, 120, 24, 6, 2, 1, 1, 40320, 5040, 720, 120, 24, 6, 2, 1, 1, 362880, 40320, 5040, 720, 120, 24, 6, 2, 1, 1, 3628800, 362880, 40320, 5040, 720, 120, 24, 6, 2, 1, 1, 39916800
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OFFSET
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0,4
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COMMENTS
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The reciprocal of each entry in a lower triangular readout of the exponential of a matrix whose entry {j+1,j} equals one (and all other entries are zero). Note all said entries are unit fractions (all numerators are one).
Multiplying the n-th diagonal by b_n with b_0 = 1 and then beheading the triangle provides a Gram matrix whose determinant is related to the reciprocal of e.g.f.s as presented in A133314. - Tom Copeland, Dec 04 2016
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LINKS
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FORMULA
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a(n) = Gamma(binomial(1 + floor((1/2) + sqrt(2*(1 + n))), 2) - n).
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EXAMPLE
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Triangle starts:
1;
1, 1;
2, 1, 1;
6, 2, 1, 1;
24, 6, 2, 1, 1;
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MATHEMATICA
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Table[Gamma[Binomial[1 + Floor[(1/2) + Sqrt[2*(1 + n)]], 2] - n], {n, 0, 77}]
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PROG
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(Magma) [[Factorial(n-k): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Jun 18 2015
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CROSSREFS
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KEYWORD
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AUTHOR
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Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006
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EXTENSIONS
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STATUS
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approved
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