Search: id:A048993
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%I A048993
%S A048993 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,
%T A048993 350,140,21,1,0,1,127,966,1701,1050,266,28,1,0,1,255,3025,7770,
%U A048993 6951,2646,462,36,1,0,1,511,9330,34105,42525,22827,5880,750,45,1
%N A048993 Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0<=k<=n.
%C A048993 Also known as Stirling set numbers. S(n,k) enumerates partitions of an 
               n-set into k non-empty subsets.
%C A048993 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_d\
               e), Oct 14 2005.
%C A048993 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:
%C A048993 B_0(x) = 1;
%C A048993 B_1(x) = x;
%C A048993 B_2(x) = x + x^2;
%C A048993 B_3(x) = x + 3x^2 + x^3;
%C A048993 B_4(x) = x + 7x^2 + 6x^3 + x^4;
%C A048993 B_5(x) = x + 15x^2 + 25x^3 + 10x^4 + x^5;
%C A048993 B_6(x) = x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6;
%D A048993 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.
%D A048993 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.
%D A048993 J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.
%D A048993 F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied 
               Tables, Cambridge, 1966, p. 223.
%D A048993 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, 
               Reading, MA, 1990, p. 244.
%D A048993 J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
%H A048993 David W. Wilson, <a href="b048993.txt">Table of n, a(n) for n = 0..10010</
               a>
%H A048993 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A048993 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling2.html">
               Stirling numbers of the second kind</a>
%H A048993 G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, <a 
               href="http://arXiv.org/abs/quant-ph/0401126">One-parameter groups 
               and combinatorial physics</a>.
%F A048993 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.
%F A048993 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.
%F A048993 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
%F A048993 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
%F A048993 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
%F A048993 Equals the inverse binomial transform of A008277 - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jan 29 2008
%F A048993 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). [From 
               Paul Barry (pbarry(AT)wit.ie), Dec 06 2009]
%e A048993 Triangle begins:
%e A048993 1
%e A048993 0 1
%e A048993 0 1 1
%e A048993 0 1 3 1
%e A048993 0 1 7 6 1
%e A048993 0 1 15 25 10 1
%e A048993 0 1 31 90 65 15 1
%e A048993 ...
%p A048993 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
%o A048993 (PARI) S(n,k) = if(k<0|k>n,0, if(n==0,1,k*S(n-1,k)+S(n-1,k-1)));
%o A048993 (PARI) for(n=0,7, for(k=0,n,print1(" "S(n,k)); if(k==n,print1(";"))))
%Y A048993 See especially A008277 which is the main entry for this triangle. Cf. 
               A008275, A039810-A039813, A048994.
%Y A048993 A000110(n) = sum(S(n, k)) k=0..n, n >= 0. Cf. A085693.
%Y A048993 Cf. A084938.
%Y A048993 Cf. A106800 (mirror image).
%Y A048993 Cf. A008277.
%Y A048993 Sequence in context: A144644 A151509 A151511 this_sequence A112413 A122960 
               A091480
%Y A048993 Adjacent sequences: A048990 A048991 A048992 this_sequence A048994 A048995 
               A048996
%K A048993 nonn,tabl,nice,new
%O A048993 0,9
%A A048993 N. J. A. Sloane (njas(AT)research.att.com).

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