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A008278 Triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1<=k<=n. 16
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; text; internal format)
OFFSET

1,5

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. Wolfdieter Lang, 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.

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

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.

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].

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015

U. N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.

Eric Weisstein's World of Mathematics, Bell Polynomial

Eric Weisstein's World of Mathematics, Empty Graph

Eric Weisstein's World of Mathematics, Independence Polynomial

Eric Weisstein's World of Mathematics, Sigma Polynomial

Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind

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.

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!).

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,

  ...

MATHEMATICA

rows = 10; Flatten[Table[StirlingS2[n, k], {n, 1, rows}, {k, n, 1, -1}]] (* Jean-Fran├žois Alcover, Nov 17 2011, *)

Table[CoefficientList[x^n BellB[n, 1/x], x], {n, 10}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)

PROG

(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

-- Reinhard Zumkeller, Jun 22 2013

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: A210866 A245474 A133713 * A213735 A056858 A137251

Adjacent sequences:  A008275 A008276 A008277 * A008279 A008280 A008281

KEYWORD

nonn,tabl,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 21 11:52 EDT 2018. Contains 313939 sequences. (Running on oeis4.)