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A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.

%I A049020 #102 Jul 02 2025 16:01:57
%S A049020 1,1,1,2,3,1,5,10,6,1,15,37,31,10,1,52,151,160,75,15,1,203,674,856,
%T A049020 520,155,21,1,877,3263,4802,3556,1400,287,28,1,4140,17007,28337,24626,
%U A049020 11991,3290,490,36,1,21147,94828,175896,174805,101031,34671,6972,786,45,1
%N A049020 Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.
%C A049020 Triangle a(n,k) read by rows; given by [1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deléham's operator defined in A084938.
%C A049020 Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. - _Paul Barry_, Jan 12 2009
%C A049020 Equal to A048993*A007318. - _Philippe Deléham_, Oct 31 2011
%C A049020 This lower unitriangular array is the L factor in the LU decomposition of the Hankel matrix (Bell(i+j-2))_i,j >= 1, where Bell(n) = A000110(n). The U factor is A059098 (see Chamberland, p. 1672). - _Peter Bala_, Oct 15 2023
%H A049020 Alois P. Heinz, <a href="/A049020/b049020.txt">Rows n = 0..140, flattened</a>
%H A049020 M. Aigner, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00108-9">A characterization of the Bell numbers</a>, Discr. Math., 205 (1999), 207-210.
%H A049020 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 A049020 Zhanar Berikkyzy, Pamela E. Harris, Anna Pun, Catherine Yan, and Chenchen Zhao, <a href="https://arxiv.org/abs/2308.14183">Combinatorial Identities for Vacillating Tableaux</a>, arXiv:2308.14183 [math.CO], 2023. See p. 11.
%H A049020 Marc Chamberland, <a href="https://doi.org/10.1016/j.laa.2011.08.030">Factored matrices can generate combinatorial identities</a>, Linear Algebra and its Applications, Volume 438, Issue 4, 15 Feb. 2013, pp. 1667-1677.
%H A049020 J. East and R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014.
%H A049020 Tom Halverson and Theodore N. Jacobson, <a href="https://arxiv.org/abs/1808.08118">Set-partition tableaux and representations of diagram algebras</a>, arXiv:1808.08118 [math.RT], 2018.
%H A049020 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 A049020 Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H A049020 W. F. Lunnon et al., <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa35/aa3511.pdf">Arithmetic properties of Bell numbers to a composite modulus I</a>, Acta Arith., 35 (1979), 1-16.
%H A049020 J. Riordan, <a href="/A001861/a001861_1.pdf">Letter, Oct 31 1977</a>. The array is on the second page.
%F A049020 a(n,k) = a(n-1, k-1) + (k+1)*a(n-1, k) + (k+1)*a(n-1, k+1), n >= 1.
%F A049020 a(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(i,k). - _Vladeta Jovovic_, Jan 27 2001
%F A049020 E.g.f. for the k-th column is (1/k!)*(exp(x)-1)^k*exp(exp(x)-1). - _Vladeta Jovovic_, Jan 27 2001
%F A049020 G.f.: 1/(1-x-xy-x^2(1+y)/(1-2x-xy-2x^2(1+y)/(1-3x-xy-3x^2(1+y)/(1-4x-xy-4x^2(1+y)/(1-... (continued fraction). - _Paul Barry_, Apr 29 2009
%F A049020 E.g.f.: exp((y+1)*(exp(x)-1)). - _Geoffrey Critzer_, Nov 30 2012
%F A049020 Note that A244489 (which is essentially the same triangle) has the formula T(n,k) = Sum_{j=k..n} binomial(n,j)*Stirling_2(j,k)*Bell(n-j), where Bell(n) = A000110(n), for n >= 1, 0 <= k <= n-1. - _N. J. A. Sloane_, May 17 2016
%F A049020 a(2n,n) = A245109(n). - _Alois P. Heinz_, Aug 23 2017
%F A049020 Empirical: a(n,k) = p(1^n)[st(1^k)] (see A002872 for notation). - _Andrey Zabolotskiy_, Oct 17 2017
%F A049020 a(n,k) = Sum_{j=0..k} (-1)^(k-j)*A046716(k, k-j)*Bell(n + j)/k!. - _Peter Luschny_, Dec 06 2023
%e A049020 Triangle begins:
%e A049020    1;
%e A049020    1,  1;
%e A049020    2,  3,  1;
%e A049020    5, 10,  6,  1;
%e A049020   15, 37, 31, 10,  1;
%e A049020   ...
%e A049020 From _Paul Barry_, Jan 12 2009: (Start)
%e A049020 Production array begins
%e A049020   1, 1;
%e A049020   1, 2, 1;
%e A049020   0, 2, 3, 1;
%e A049020   0, 0, 3, 4, 1;
%e A049020   0, 0, 0, 4, 5, 1;
%e A049020   ... (End)
%p A049020 a:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A049020       `if`(n=0, 1, a(n-1, k-1) +(k+1)*(a(n-1, k) +a(n-1, k+1))))
%p A049020     end:
%p A049020 seq(seq(a(n, k), k=0..n), n=0..15);  # _Alois P. Heinz_, Nov 30 2012
%t A049020 a[n_, k_] = Sum[StirlingS2[n, i]*Binomial[i, k], {i, 0, n}]; Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]]
%t A049020 (* _Jean-François Alcover_, Aug 29 2011, after _Vladeta Jovovic_ *)
%o A049020 (PARI) T(n,k)=if(k<0 || k>n,0,n!*polcoeff(polcoeff(exp((1+y)*(exp(x+x*O(x^n))-1)),n),k))
%o A049020 (Sage)
%o A049020 def A049020_triangle(dim):
%o A049020     M = matrix(ZZ, dim, dim)
%o A049020     for n in (0..dim-1): M[n, n] = 1
%o A049020     for n in (1..dim-1):
%o A049020         for k in (0..n-1):
%o A049020             M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k]+(k+1)*M[n-1, k+1]
%o A049020     return M
%o A049020 A049020_triangle(9) # _Peter Luschny_, Sep 19 2012
%Y A049020 First column gives A000110, second column = A005493.
%Y A049020 Third column = A003128, row sums = A001861, A059340.
%Y A049020 See A244489 for another version of this triangle.
%Y A049020 Cf. A059098, A245109, A046716.
%K A049020 nonn,tabl,nice,easy
%O A049020 0,4
%A A049020 _N. J. A. Sloane_
%E A049020 More terms from _James Sellers_.
%E A049020 Better definition from _Geoffrey Critzer_, Nov 30 2012.