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%I #18 Sep 08 2022 08:45:25
%S 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,
%T 5040,720,120,24,6,2,1,1,40320,5040,720,120,24,6,2,1,1,362880,40320,
%U 5040,720,120,24,6,2,1,1,3628800,362880,40320,5040,720,120,24,6,2,1,1,39916800
%N Triangle read by rows, T(n,k) = (n-k)!, for n>=0 and 0<=k<=n.
%C 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).
%C Denominators of unfinished fractional coefficients for polynomials A152650/A152656 = A009998/A119052. - _Paul Curtz_, Dec 13 2008
%C 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
%F T(n,k) = A025581(n,k)!.
%F a(n) = Gamma(binomial(1 + floor((1/2) + sqrt(2*(1 + n))), 2) - n).
%e Triangle starts:
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 6, 2, 1, 1;
%e 24, 6, 2, 1, 1;
%t Table[Gamma[Binomial[1 + Floor[(1/2) + Sqrt[2*(1 + n)]], 2] - n], {n, 0, 77}]
%o (Magma) [[Factorial(n-k): k in [1..n]]: n in [1.. 15]]; // _Vincenzo Librandi_, Jun 18 2015
%Y Cf. A025581.
%Y Cf. A133314.
%K easy,nonn,tabl
%O 0,4
%A Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2006
%E Name edited by _Peter Luschny_, Jun 17 2015
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