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 A202917 For n >= 0, let n!^(1) = A053657(n+1) and, for 0 <= m <= n, C^(1)(n,m) = n!^(1)/(m!^(1)*(n-m)!^(1)). The sequence gives a triangle of numbers C^(1)(n,m) with rows of length n+1. 10
 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 1, 60, 10, 60, 1, 1, 1, 10, 10, 1, 1, 1, 126, 21, 1260, 21, 126, 1, 1, 1, 21, 21, 21, 21, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS 1) Note that A053657(n+1) is the LCM of the denominators of the coefficients of the polynomials Q^(1)_n(x) which, for integer x=k, are defined by the recursion Q^(1)_0(x)=1, for n>=1, Q^(1)_n(x) = Sum_{i=1..k} i*Q^(1)_(n-1)(i). Also note that Q^(1)_n(k) = S(k+n,k), where the numbers S(l,m) are Stirling numbers of the second kind. The sequence of polynomials {Q^(1)_n(x)} includes the family of sequences of polynomials {{Q^(r)_n}}_(r>=0) described in a comment at A175669. In particular, the LCM of the denominators of the coefficients of Q^(0)_n(x) is n!. 2) This triangle differs from triangle A186430 which is defined according to the theory of factorials over sets by Bhargava. Unfortunately, this theory does not have a conversion theorem. Therefore it is not known if there is a set A such that n!^(1) = n!_A in the Bhargava sense. 3) If p is an odd prime, then the (p-1)-th row contains two 1's and p-2 numbers that are multiples of p. For a conjectural generalization, see comment in A175669. LINKS FORMULA A007814(C^(1)(n,m)) = A007814(C(n,m)). EXAMPLE Triangle begins n/m.|..0.....1.....2.....3.....4.....5.....6.....7 ================================================== .0..|..1 .1..|..1.....1 .2..|..1.....6.....1 .3..|..1.....1 ... 1  .....1 .4..|..1....60....10......60.....1 .5..|..1.....1....10......10.....1.....1 .6..|..1...126....21....1260....21...126.....1 .7..|..1.....1....21......21....21....21.....1.....1 .8..| MATHEMATICA A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k, 0, n}], {p, Prime[Range[n]]}]; f1[n_] := A053657[n+1]; C1[n_, m_] := f1[n]/(f1[m] * f1[n-m]); Table[C1[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Nov 22 2016 *) CROSSREFS Cf. A175669, A053657, A202339, A202367, A202368, A202369 Sequence in context: A257936 A143532 A267426 * A324396 A234959 A325471 Adjacent sequences:  A202914 A202915 A202916 * A202918 A202919 A202920 KEYWORD nonn,tabl AUTHOR Vladimir Shevelev and Peter J. C. Moses, Dec 26 2011 STATUS approved

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Last modified May 14 15:03 EDT 2021. Contains 343884 sequences. (Running on oeis4.)