0,9

Also known as Stirling set numbers. S(n,k) enumerates partitions of an n-set into k non-empty 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. W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Oct 14 2005.

Comment from Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 14 2007: 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) = x;

B_2(x) = x + x^2;

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

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

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

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

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.

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].

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.

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 Deleham's operator defined in A084938.

Sum_{k = 0..n} x^k*S(n, k) = A000110(n), A001861(n), A027710(n), A078944(n) for x = 1, 2, 3, 4 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), May 09 2004

S(n, k)=sum{i=0..k, (-1)^(k+i)binomial(k, i)i^n/k!} - Paul Barry (pbarry(AT)wit.ie), 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 (deutsch(AT)duke.poly.edu), Nov 01 2006

Equals the inverse binomial transform of A008277 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 29 2008

Triangle begins:

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

...

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

(PARI) S(n, k) = if(k<0|k>n, 0, if(n==0, 1, k*S(n-1, k)+S(n-1, k-1)));

(PARI) for(n=0, 7, for(k=0, n, print1(" "S(n, k)); if(k==n, print1("; "))))

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.

Cf. A106800 (mirror image).

Cf. A008277.

Sequence in context: A144644 A151509 A151511 this_sequence A112413 A122960 A091480

Adjacent sequences: A048990 A048991 A048992 this_sequence A048994 A048995 A048996

nonn,tabl,nice

N. J. A. Sloane (njas(AT)research.att.com).