|
%I #84 Mar 27 2021 08:09:08
%S 1,1,1,1,3,1,1,6,7,1,1,10,25,15,1,1,15,65,90,31,1,1,21,140,350,301,63,
%T 1,1,28,266,1050,1701,966,127,1,1,36,462,2646,6951,7770,3025,255,1,1,
%U 45,750,5880,22827,42525,34105,9330,511,1
%N Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
%C The n-th row also gives the coefficients of the sigma polynomial of the empty graph \bar K_n. - _Eric W. Weisstein_, Apr 07 2017
%C The n-th row also gives the coefficients of the independence polynomial of the (n-1)-triangular honeycomb bishop graph. - _Eric W. Weisstein_, Apr 03 2018
%C From _Gus Wiseman_, Aug 11 2020: (Start)
%C Conjecture: also the number of divisors of the superprimorial A006939(n - 1) that have 0 <= k <= n distinct prime factors, all appearing with distinct multiplicities. For example, row n = 4 counts the following divisors of 360:
%C 1 2 12 360
%C 3 18
%C 4 20
%C 5 24
%C 8 40
%C 9 45
%C 72
%C Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i with k nonzero values, all of which are distinct.
%C Crossrefs:
%C A006939 lists superprimorials or Chernoff numbers.
%C A022915 counts permutations of prime indices of superprimorials.
%C A076954 can be used instead of A006939.
%C A130091 lists numbers with distinct prime multiplicities.
%C A181796 counts divisors with distinct prime multiplicities.
%C A336420 is the version counting all prime factors, not just distinct ones.
%C Cf. A000005, A027423, A095149, A124010, A317829, A327498, A336419, A336421.
%C (End)
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
%D F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994.
%H T. D. Noe, <a href="/A008278/b008278.txt">Rows n = 0..100 of triangle, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Noureddine Chair, <a href="https://doi.org/10.1016/j.aop.2012.09.002">Exact two-point resistance, and the simple random walk on the complete graph minus N edges</a>, Ann. Phys. 327, No. 12, 3116-3129 (2012), eq. (27).
%H Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, and Alan D. Sokal, <a href="https://arxiv.org/abs/2012.03629">Coefficientwise total positivity of some matrices defined by linear recurrences</a>, arXiv:2012.03629 [math.CO], 2020.
%H T. Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015
%H U. N. Katugampola, <a href="http://arxiv.org/abs/1106.0965">A new Fractional Derivative and its Mellin Transform</a>, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EmptyGraph.html">Empty Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IndependencePolynomial.html">Independence Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SigmaPolynomial.html">Sigma Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling Number of the Second Kind</a>
%F T(n, k)=0 if n < k, T(n, 0)=0, T(1, 1)=1, T(n, k) = (n-k+1)*T(n-1, k-1) + T(n-1, k) otherwise.
%F O.g.f. for the k-th column: 1/(1-x) if k=1 and A(k,x):=((x^k)/(1-x)^(2*k+1))*Sum_{m=0..k-1} A008517(k,m+1)*x^m if k >= 2. A008517 is the second-order Eulerian triangle. Cf. p. 257, eq. (6.43) of the R. L. Graham et al. book. - _Wolfdieter Lang_, Oct 14 2005
%F E.g.f. for the k-th column (with offset n=0): E(k,x):=exp(x)*Sum_{m=0..k-1} A112493(k-1,m)*(x^(k-1+m))/(k-1+m)! if k >= 1. - _Wolfdieter Lang_, Oct 14 2005
%F a(n) = abs(A213735(n-1)). - _Hugo Pfoertner_, Sep 07 2020
%e The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).
%e Triangle starts:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 6, 7, 1;
%e 1, 10, 25, 15, 1;
%e 1, 15, 65, 90, 31, 1;
%e 1, 21, 140, 350, 301, 63, 1;
%e 1, 28, 266, 1050, 1701, 966, 127, 1;
%e 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1;
%e ...
%t rows = 10; Flatten[Table[StirlingS2[n, k], {n, 1, rows}, {k, n, 1, -1}]] (* _Jean-François Alcover_, Nov 17 2011, *)
%t Table[CoefficientList[x^n BellB[n, 1/x], x], {n, 10}] // Flatten (* _Eric W. Weisstein_, Apr 05 2017 *)
%o (Haskell)
%o a008278 n k = a008278_tabl !! (n-1) !! (k-1)
%o a008278_row n = a008278_tabl !! (n-1)
%o a008278_tabl = iterate st2 [1] where
%o st2 row = zipWith (+) ([0] ++ row') (row ++ [0])
%o where row' = reverse $ zipWith (*) [1..] $ reverse row
%o -- _Reinhard Zumkeller_, Jun 22 2013
%o (PARI) for(n=1,10,for(k=1,n,print1(stirling(n,n-k+1,2),", "))) \\ _Hugo Pfoertner_, Aug 30 2020
%Y See A008277 and A048993, which are the main entries for this triangle of numbers.
%Y Cf. A094262, A008277, A008276, A003422, A000166, A000110, A000204, A000045, A000108, A213735.
%K nonn,tabl,nice
%O 1,5
%A _N. J. A. Sloane_
%E Name edited by _Gus Wiseman_, Aug 11 2020
|
