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A008278
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Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.
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23
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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, 1, 1, 28, 266, 1050, 1701, 966, 127, 1, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1
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OFFSET
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1,5
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COMMENTS
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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
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
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:
1 2 12 360
3 18
4 20
5 24
8 40
9 45
72
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i with k nonzero values, all of which are distinct.
Crossrefs:
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336420 is the version counting all prime factors, not just distinct ones.
(End)
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REFERENCES
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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.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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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.
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
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
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EXAMPLE
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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!).
Triangle starts:
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, 1;
1, 28, 266, 1050, 1701, 966, 127, 1;
1, 36, 462, 2646, 6951, 7770, 3025, 255, 1;
...
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MATHEMATICA
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rows = 10; Flatten[Table[StirlingS2[n, k], {n, 1, rows}, {k, n, 1, -1}]] (* Jean-François Alcover, Nov 17 2011, *)
Table[CoefficientList[x^n BellB[n, 1/x], x], {n, 10}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
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PROG
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(Haskell)
a008278 n k = a008278_tabl !! (n-1) !! (k-1)
a008278_row n = a008278_tabl !! (n-1)
a008278_tabl = iterate st2 [1] where
st2 row = zipWith (+) ([0] ++ row') (row ++ [0])
where row' = reverse $ zipWith (*) [1..] $ reverse row
(PARI) for(n=1, 10, for(k=1, n, print1(stirling(n, n-k+1, 2), ", "))) \\ Hugo Pfoertner, Aug 30 2020
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CROSSREFS
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See A008277 and A048993, which are the main entries for this triangle of numbers.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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