This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A048993 #180 May 01 2025 08:29:34
%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,
%T A048993 63,301,350,140,21,1,0,1,127,966,1701,1050,266,28,1,0,1,255,3025,7770,
%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.
%C A048993 S(n,k) enumerates partitions of an n-set into k nonempty 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_{m=0..k-1} A008517(k,m+1)*x^m. A008517 is the second-order Eulerian triangle. - _Wolfdieter Lang_, Oct 14 2005
%C A048993 From _Philippe Deléham_, Nov 14 2007: (Start)
%C A048993 Sum_{k=0..n} S(n,k)*x^k = B_n(x), where B_n(x) = Bell polynomials.
%C A048993 The first few Bell polynomials are:
%C A048993 B_0(x) = 1;
%C A048993 B_1(x) = 0 + x;
%C A048993 B_2(x) = 0 + x + x^2;
%C A048993 B_3(x) = 0 + x + 3x^2 + x^3;
%C A048993 B_4(x) = 0 + x + 7x^2 + 6x^3 + x^4;
%C A048993 B_5(x) = 0 + x + 15x^2 + 25x^3 + 10x^4 + x^5;
%C A048993 B_6(x) = 0 + x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6;
%C A048993 (End)
%C A048993 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
%C A048993 Also the inverse Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265604. - _Peter Luschny_, Dec 31 2015
%C A048993 From _Wolfdieter Lang_, Feb 21 2017: (Start)
%C A048993 The transposed (trans) of this lower triagonal Sheffer matrix of the associated type S = (1, exp(x) - 1) (taken as N X N matrix for arbitrarily large N) provides the transition matrix from the basis {x^n/n!}, n >= 0, to the basis {y^n/n!}, n >= 0, with y^n/n! = Sum_{m>=n} S^{trans}(n, m) x^m/m! = Sum_{m>=0} x^m/m!*S(m, n).
%C A048993 The Sheffer transform with S = (g, f) of a sequence {a_n} to {b_n} for n >= 0, in matrix notation vec(b) = S vec(a), satisfies, with e.g.f.s A and B, B(x) = g(x)*A(f(x)) and B(x) = A(y(x)) identically, with vec(xhat) = S^{trans,-1} vec(yhat) in symbolic notation with vec(xhat)_n = x^n/n! (similarly for vec(yhat)).
%C A048993 (End)
%C A048993 For k >= 1 S(n, k) = h^{(k)}_{n-k}, the complete homogeneous symmetric function of the k symbols 1,2, ..., k, of degree n-k. Thus S(n, k) is for k >= 1 the (dimensionless) volume of the multichoose(k, n-k) = binomial(n-1, k-1) polytopes of dimension n-k with (dimensionless) side lengths from the set {1, 2, ..., k}. See an example below. - _Wolfdieter Lang_, May 26 2017
%C A048993 Number of partitions of {1, 2, ..., n+1} into k+1 nonempty subsets such that no subset contains two adjacent numbers. - _Thomas Anton_, Sep 26 2022
%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="/A048993/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.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A048993 V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.
%H A048993 Peter Bala, <a href="/A048993/a048993.pdf">The white diamond product of power series</a>
%H A048993 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry1/barry263.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations</a>, Journal of Integer Sequences, 17 (2014), #14.2.3.
%H A048993 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Barry2/barry281.html">Constructing Exponential Riordan Arrays from Their A and Z Sequences</a>, Journal of Integer Sequences, 17 (2014), #14.2.6.
%H A048993 Paul Barry, <a href="https://arxiv.org/abs/1803.06408">Three Études on a sequence transformation pipeline</a>, arXiv:1803.06408 [math.CO], 2018.
%H A048993 Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, and Alan D. Sokal, <a href="https://arxiv.org/abs/2012.03629">Coefficientwise total positivity of some matrices defined by linear recurrences</a>, arXiv:2012.03629 [math.CO], 2020.
%H A048993 R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling2.html">Stirling numbers of the second kind</a>
%H A048993 Gerard Duchamp, Karol A. Penson, Allan I. Solomon, Andrej Horzela, and Pawel Blasiak, <a href="http://arXiv.org/abs/quant-ph/0401126">One-parameter groups and combinatorial physics</a>, arXiv:quant-ph/0401126, 2004.
%H A048993 FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/StatisticsDatabase/St000105">The number of blocks in the set partition.</a>
%H A048993 Bill Gosper, <a href="/A008277/a008277.png">Colored illustrations of triangle of Stirling numbers of second kind read mod 2, 3, 4, 5, 6, 7</a>
%H A048993 W. Steven Gray and Makhin Thitsa, <a href="http://dx.doi.org/10.1109/SSST.2013.6524939">System Interconnections and Combinatorial Integer Sequences</a>, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
%H A048993 Aoife Hennessy and Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry6/barry161.html">Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials</a>, J. Int. Seq. 14 (2011) # 11.8.2.
%H A048993 Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See pp. 8-9.
%H A048993 Mathias Pétréolle and Alan D. Sokal, <a href="https://arxiv.org/abs/1907.02645">Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions</a>, arXiv:1907.02645 [math.CO], 2019.
%H A048993 Claus Michael Ringel, <a href="http://arxiv.org/abs/1502.06553">The Catalan combinatorics of the hereditary artin algebras</a>, arXiv preprint arXiv:1502.06553 [math.RT], 2015.
%H A048993 X.-T. Su, D.-Y. Yang, and W.-W. Zhang, <a href="http://ajc.maths.uq.edu.au/pdf/56/ajc_v56_p133.pdf">A note on the generalized factorial</a>, Australasian Journal of Combinatorics, Volume 56 (2013), Pages 133-137.
%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 Deléham's operator defined in A084938.
%F A048993 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
%F A048993 S(n, k) = Sum_{i=0..k} (-1)^(k+i)binomial(k, i)i^n/k!. - _Paul Barry_, Aug 05 2004
%F A048993 Sum_{k=0..n} k*S(n,k) = B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - _Emeric Deutsch_, Nov 01 2006
%F A048993 Equals the inverse binomial transform of A008277. - _Gary W. Adamson_, 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 Deléham DELTA construction). - _Paul Barry_, Dec 06 2009
%F A048993 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
%F A048993 Inverse of padded A008275 (padded just as A048993 = padded A008277). - _Tom Copeland_, Apr 25 2014
%F A048993 E.g.f. for the row polynomials s(n,x) = Sum_{k=0..n} S(n,k)*x^k 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
%F A048993 G.f. for column k: x^k/Product_{j=1..k} (1-j*x), k >= 0 (with the empty product for k = 0 put to 1). See Abramowitz-Stegun, p. 824, 24.1.4 B. - _Wolfdieter Lang_, May 26 2017
%F A048993 Boas-Buck recurrence for column sequence m: S(n, k) = (k/(n - k))*(n*S(n-1, k)/2 + Sum_{p=k..n-2} (-1)^(n-p)*binomial(n,p)*Bernoulli(n-p)*S(p, k)), for n > k >= 0, with input T(k,k) = 1. See a comment and references in A282629. An example is given below. - _Wolfdieter Lang_, Aug 11 2017
%F A048993 The n-th row polynomial has the form x o x o ... o x (n factors), where o denotes the white diamond multiplication operator defined in Bala - see Example E4. - _Peter Bala_, Jan 07 2018
%F A048993 Sum_{k=1..n} k*S(n,k) = A138378(n). - _Alois P. Heinz_, Jan 07 2022
%F A048993 S(n,k) = Sum_{j=k..n} (-1)^(j-k)*A059297(n,j)*A354794(j,k). - _Mélika Tebni_, Jan 27 2023
%e A048993 The triangle S(n,k) begins:
%e A048993 n\k 0 1 2 3 4 5 6 7 8 9 10 11 12
%e A048993 0: 1
%e A048993 1: 0 1
%e A048993 2: 0 1 1
%e A048993 3: 0 1 3 1
%e A048993 4: 0 1 7 6 1
%e A048993 5: 0 1 15 25 10 1
%e A048993 6: 0 1 31 90 65 15 1
%e A048993 7: 0 1 63 301 350 140 21 1
%e A048993 8: 0 1 127 966 1701 1050 266 28 1
%e A048993 9: 0 1 255 3025 7770 6951 2646 462 36 1
%e A048993 10: 0 1 511 9330 34105 42525 22827 5880 750 45 1
%e A048993 11: 0 1 1023 28501 145750 246730 179487 63987 11880 1155 55 1
%e A048993 12: 0 1 2047 86526 611501 1379400 1323652 627396 159027 22275 1705 66 1
%e A048993 ... reformatted and extended - _Wolfdieter Lang_, Oct 16 2014
%e A048993 Completely symmetric function S(4, 2) = h^{(2)}_2 = 1^2 + 2^2 + 1^1*2^1 = 7; S(5, 2) = h^{(2)}_3 = 1^3 + 2^3 + 1^2*2^1 + 1^1*2^2 = 15. - _Wolfdieter Lang_, May 26 2017
%e A048993 From _Wolfdieter Lang_, Aug 11 2017: (Start)
%e A048993 Recurrence: S(5, 3) = S(4, 2) + 2*S(4, 3) = 7 + 3*6 = 25.
%e A048993 Boas-Buck recurrence for column m = 3, and n = 5: S(5, 3) = (3/2)*((5/2)*S(4, 3) + 10*Bernoulli(2)*S(3, 3)) = (3/2)*(15 + 10*(1/6)*1) = 25. (End)
%p A048993 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
%t A048993 t[n_, k_] := StirlingS2[n, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Robert G. Wilson v_ *)
%o A048993 (PARI) for(n=0, 22, for(k=0, n, print1(stirling(n, k, 2), ", ")); print()); \\ _Joerg Arndt_, Apr 21 2013
%o A048993 (Maxima) create_list(stirling2(n,k),n,0,12,k,0,n); /* _Emanuele Munarini_, Mar 11 2011 */
%o A048993 (Haskell)
%o A048993 a048993 n k = a048993_tabl !! n !! k
%o A048993 a048993_row n = a048993_tabl !! n
%o A048993 a048993_tabl = iterate (\row ->
%o A048993 [0] ++ (zipWith (+) row $ zipWith (*) [1..] $ tail row) ++ [1]) [1]
%o A048993 -- _Reinhard Zumkeller_, Mar 26 2012
%Y A048993 See especially A008277 which is the main entry for this triangle.
%Y A048993 Cf. A008275, A039810-A039813, A048994.
%Y A048993 A000110(n) = sum(S(n, k)) k=0..n, n >= 0. Cf. A085693.
%Y A048993 Cf. A084938, A106800 (mirror image), A138378, A213061 (mod 2).
%Y A048993 Cf. A059297, A354794.
%K A048993 nonn,tabl,nice
%O A048993 0,9
%A A048993 _N. J. A. Sloane_, Dec 11 1999