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A008278
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Triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1<=k<=n.
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14
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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
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| O.g.f. for the k-th column sequence is 1/(1-x) if k=1 and A(k,x):=((x^k)/(1-x)^(2*k+1))*sum(A008517(k,m+1)*x^m,m=0..k-1) if k>=2. A008517 is the second-order Eulerian triangle. Cf. p. 257, eq. (6.43) of the R. L. Graham et al. book. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 14 2005.
E.g.f. for the k-th column sequence with offset n=0 is E(k,x):= exp(x)*sum(A112493(k-1,m)*(x^(k-1+m))/(k-1+m)!,m=0..k-1) if k>=1. W. Lang, Oct 14 2005.
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REFERENCES
| 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.
U. N. Katugampola, A new Fractional Derivative and its Mellin Transform, Arxiv preprint arXiv:1106.0965, 2011
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LINKS
| T. D. Noe, Rows n=0..100 of triangle, flattened
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
| a(n, k)=0 if n<k, a(n, 0)=0, a(1, 1)=1, a(n, k)= (n-k+1)*a(n-1, k-1) + a(n-1, k) else.
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EXAMPLE
| 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!).
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MATHEMATICA
| rows = 10; Flatten[ Table[ StirlingS2[n, k], {n, 1, rows}, {k, n, 1, -1}]] (* From Jean-François Alcover, Nov 17 2011, *)
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CROSSREFS
| See A008277 and A048993, which are the main entries for this triangle of numbers.
Cf. A094262, A008277, A008276, A003422, A000166, A000110, A000204, A000045, A000108.
Sequence in context: A143362 A182823 A133713 * A056858 A137251 A158359
Adjacent sequences: A008275 A008276 A008277 * A008279 A008280 A008281
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KEYWORD
| nonn,tabl,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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