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A048993 Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0<=k<=n. 109
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 7, 6, 1, 0, 1, 15, 25, 10, 1, 0, 1, 31, 90, 65, 15, 1, 0, 1, 63, 301, 350, 140, 21, 1, 0, 1, 127, 966, 1701, 1050, 266, 28, 1, 0, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 0, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

Also known as Stirling set numbers.

S(n,k) enumerates partitions of an n-set into k nonempty subsets.

The o.g.f. for the sequence of diagonal k (k=0 for the main diagonal) is G(k,x)= ((x^k)/(1-x)^(2*k+1))*sum(A008517(k,m+1)*x^m,m=0..k-1). A008517 is the second-order Eulerian triangle. - Wolfdieter Lang, Oct 14 2005.

From Philippe Deléham, Nov 14 2007: (Start)

Sum_{k, 0<=k<=n}S(n,k)*x^k = B_n(x), where B_n(x) = Bell polynomials. The first few Bell polynomials are:

B_0(x) = 1;

B_1(x) = 0 + x;

B_2(x) = 0 + x + x^2;

B_3(x) = 0 + x + 3x^2 + x^3;

B_4(x) = 0 + x + 7x^2 + 6x^3 + x^4;

B_5(x) = 0 + x + 15x^2 + 25x^3 + 10x^4 + x^5;

B_6(x) = 0 + x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6;

(End)

This is the Sheffer triangle (1, exp(x) - 1), an exponential (binomial) convolution triangle. The a-sequence is given by A006232/A006233 (Cauchy sequence). The z-sequence is the zero sequence. See the link under A006232 for the definition and use of these sequences. The row sums give A000110 (Bell), and the alternating row sums give A000587 (see the Philippe Deléham formulas and crossreferences below). - Wolfdieter Lang, Oct 16 2014

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.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.

J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.

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, Reading, MA, 1990, p. 244.

J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

LINKS

David W. Wilson, Table of n, a(n) for n = 0..10010

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].

P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.

P. Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.

R. M. Dickau, Stirling numbers of the second kind

G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics.

W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.

X.-T. Su, D.-Y. Yang, W.-W. Zhang, A note on the generalized factorial, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.

FORMULA

S(n, k) = k*S(n-1, k) + S(n-1, k-1), n>0; S(0, k) = 0, k>0; S(0, 0)=1.

Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ..] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.

Sum_{k = 0..n} x^k*S(n, k) = A213170(n), A000587(n), A000007(n), A000110(n), A001861(n), A027710(n), A078944(n), A144180(n), A144223(n), A144263(n) respectively for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7. - Philippe Deléham, May 09 2004, Feb 16 2013

S(n, k)=sum{i=0..k, (-1)^(k+i)binomial(k, i)i^n/k!}. - Paul Barry, Aug 05 2004

Sum(k*S(n,k), k=0..n)=B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - Emeric Deutsch, Nov 01 2006

Equals the inverse binomial transform of A008277. - Gary W. Adamson, Jan 29 2008

G.f.: 1/(1-xy/(1-x/(1-xy/(1-2x/(1-xy/1-3x/(1-xy/(1-4x/(1-xy/(1-5x/(1-... (continued fraction equivalent to Deleham DELTA construction). - Paul Barry, Dec 06 2009

G.f.: 1/Q(0), where Q(k) = 1 -(y+k)*x - (k+1)*y*x^2/Q(k+1) ; (continued fraction). - Sergei N. Gladkovskii, Nov 09 2013

Inverse of padded A008275 (padded just as A048993 = padded A008277). - Tom Copeland, Apr 25 2014

E.g.f. for the row polynomials s(n,x) = sum(S(n,k)*x^k, k=0..n) is exp(x*(exp(z)-1)) (Sheffer property). E.g.f. for the k-th column sequence with k leading zeros is ((exp(x)-1)^k)/k! (Sheffer property). - Wolfdieter Lang, Oct 16 2014

EXAMPLE

The triangle S(n,k) begins:

n\k  0 1    2      3       4       5       6       7       8      9    10   11 12 13 ...

0:   1

1:   0 1

2:   0 1    1

3:   0 1    3      1

4:   0 1    7      6       1

5:   0 1   15     25      10       1

6:   0 1   31     90      65      15       1

7:   0 1   63    301     350     140      21       1

8:   0 1  127    966    1701    1050     266      28       1

9:   0 1  255   3025    7770    6951    2646     462      36      1

10:  0 1  511   9330   34105   42525   22827    5880     750     45     1

11:  0 1 1023  28501  145750  246730  179487   63987   11880   1155    55    1

12:  0 1 2047  86526  611501 1379400 1323652  627396  159027  22275  1705   66  1

13:  0 1 4095 261625 2532530 7508501 9321312 5715424 1899612 359502 39325 2431 78  1

... reformatted and extended - Wolfdieter Lang, Oct 16 2014

----------------------------------------------------------------------------------------

MAPLE

with(combinat): for n from 0 to 10 do seq(stirling2(n, k), k=0..n) od; # yields sequence in triangular form - Emeric Deutsch, Nov 01 2006

MATHEMATICA

t[n_, k_] := StirlingS2[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Robert G. Wilson v *)

PROG

(PARI) for(n=0, 22, for(k=0, n, print1(stirling(n, k, 2), ", ")); print()); \\ Joerg Arndt, Apr 21 2013

(Maxima) create_list(stirling2(n, k), n, 0, 12, k, 0, n); /* Emanuele Munarini, Mar 11 2011 */

(Haskell)

a048993 n k = a048993_tabl !! n !! k

a048993_row n = a048993_tabl !! n

a048993_tabl = iterate (\row ->

   [0] ++ (zipWith (+) row $ zipWith (*) [1..] $ tail row) ++ [1]) [1]

-- Reinhard Zumkeller, Mar 26 2012

CROSSREFS

See especially A008277 which is the main entry for this triangle.

Cf. A008275, A039810-A039813, A048994.

A000110(n) = sum(S(n, k)) k=0..n, n >= 0. Cf. A085693.

Cf. A084938, A106800 (mirror image), A213061.

Sequence in context: A144644 A151509 A151511 * A193357 A112413 A122960

Adjacent sequences:  A048990 A048991 A048992 * A048994 A048995 A048996

KEYWORD

nonn,tabl,nice

AUTHOR

N. J. A. Sloane, Dec 11 1999

STATUS

approved

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Last modified November 1 00:43 EDT 2014. Contains 248876 sequences.