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 A106800 #60 Aug 03 2025 04:53:49
%S A106800 1,1,0,1,1,0,1,3,1,0,1,6,7,1,0,1,10,25,15,1,0,1,15,65,90,31,1,0,1,21,
%T A106800 140,350,301,63,1,0,1,28,266,1050,1701,966,127,1,0,1,36,462,2646,6951,
%U A106800 7770,3025,255,1,0,1,45,750,5880,22827,42525,34105,9330,511,1,0
%N A106800 Triangle of Stirling numbers of 2nd kind, S(n, n-k), n >= 0, 0 <= k <= n.
%C A106800 Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...] DELTA [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, May 19 2005
%D A106800 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 A106800 F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D A106800 Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, table 2.14.1 at page 24.
%H A106800 G. C. Greubel, <a href="/A106800/b106800.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A106800 A. Aboud, J.-P. Bultel, A. Chouria, J.-G. Luque, and O. Mallet, <a href="http://arxiv.org/abs/1402.2960">Bell polynomials in combinatorial Hopf algebras</a>, arXiv preprint arXiv:1402.2960 [math.CO], 2014.
%H A106800 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 A106800 J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013.
%H A106800 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A106800 A(x;t) = exp(t*(exp(x)-1)) = Sum_{n>=0} P_n(t) * x^n/n!, where P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k). - _Gheorghe Coserea_, Jan 30 2017
%F A106800 Also, P_n(t) * exp(t) = (t * d/dt)^n exp(t). - _Michael Somos_, Aug 16 2017
%F A106800 T(n, k) = Sum_{j=0..k} E2(k, j)*binomial(n + k - j, 2*k), where E2(k, j) are the second-order Eulerian numbers A340556. - _Peter Luschny_, Feb 21 2021
%e A106800 From _Gheorghe Coserea_, Jan 30 2017: (Start)
%e A106800 Triangle starts:
%e A106800 n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]
%e A106800 [0] 1;
%e A106800 [1] 1, 0;
%e A106800 [2] 1, 1, 0;
%e A106800 [3] 1, 3, 1, 0;
%e A106800 [4] 1, 6, 7, 1, 0;
%e A106800 [5] 1, 10, 25, 15, 1, 0;
%e A106800 [6] 1, 15, 65, 90, 31, 1, 0;
%e A106800 [7] 1, 21, 140, 350, 301, 63, 1, 0;
%e A106800 [8] 1, 28, 266, 1050, 1701, 966, 127, 1, 0;
%e A106800 [9] 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 0;
%e A106800 ...
%e A106800 (End)
%p A106800 seq(seq(Stirling2(n, n-k), k=0..n), n=0..8); # _Peter Luschny_, Feb 21 2021
%t A106800 Table[ StirlingS2[n, m], {n, 0, 10}, {m, n, 0, -1}]//Flatten (* _Robert G. Wilson v_, Jan 30 2017 *)
%o A106800 (PARI)
%o A106800 N=11; x='x+O('x^N); t='t; concat(apply(p->Vec(p), Vec(serlaplace(exp(t*(exp(x)-1)))))) \\ _Gheorghe Coserea_, Jan 30 2017
%o A106800 {T(n, k) = my(A, B); if( n<0 || k>n, 0, A = B = exp(x + x * O(x^n)); for(i=1, n, A = x * A'); polcoeff(A / B, n-k))}; /* _Michael Somos_, Aug 16 2017 */
%o A106800 (Sage) flatten([[stirling_number2(n, n-k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Sep 11 2021
%Y A106800 See A008277 and A048993, which are the main entries for this triangle of numbers.
%Y A106800 The Stirling1 counterpart is A054654.
%Y A106800 Row sum: A000110.
%Y A106800 Column 0: A000012.
%Y A106800 Column 1: A000217.
%Y A106800 Main Diagonal: A000007.
%Y A106800 1st minor diagonal: A000012.
%Y A106800 2nd minor diagonal: A000225.
%Y A106800 3rd minor diagonal: A000392.
%Y A106800 Cf. A008278, A340556.
%K A106800 nonn,tabl
%O A106800 0,8
%A A106800 _N. J. A. Sloane_