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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 18
 1, 1, 1, 2, 3, 1, 5, 10, 6, 1, 15, 37, 31, 10, 1, 52, 151, 160, 75, 15, 1, 203, 674, 856, 520, 155, 21, 1, 877, 3263, 4802, 3556, 1400, 287, 28, 1, 4140, 17007, 28337, 24626, 11991, 3290, 490, 36, 1, 21147, 94828, 175896, 174805, 101031, 34671, 6972, 786, 45, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS 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. Exponential Riordan array [exp(exp(x)-1), exp(x)-1]. - Paul Barry, Jan 12 2009 Equal to A048993*A007318. - Philippe Deléham, Oct 31 2011 LINKS Alois P. Heinz, Rows n = 0..140, flattened M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210. P. Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6. J. East, R. D. Gray, Idempotent generators in finite partition monoids and related semigroups, arXiv preprint arXiv:1404.2359 [math.GR], 2014. Tom Halverson, Theodore N. Jacobson, Set-partition tableaux and representations of diagram algebras, arXiv:1808.08118 [math.RT], 2018. A. Hennessy, P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2. Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020. W. F. Lunnon et al., Arithmetic properties of Bell numbers to a composite modulus I, Acta Arith., 35 (1979), 1-16. J. Riordan, Letter, Oct 31 1977. The array is on the second page. FORMULA a(n,k) = a(n-1, k-1) + (k+1)*a(n-1, k) + (k+1)*a(n-1, k+1), n >= 1. a(n,k) = Sum_{i=0..n} Stirling2(n,i)*binomial(i,k). - Vladeta Jovovic, Jan 27 2001 E.g.f. for the k-th column is (1/k!)*(exp(x)-1)^k*exp(exp(x)-1). - Vladeta Jovovic, Jan 27 2001 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 E.g.f.: exp((y+1)*(exp(x)-1)). - Geoffrey Critzer, Nov 30 2012 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 a(2n,n) = A245109(n). - Alois P. Heinz, Aug 23 2017 Empirical: a(n,k) = p(1^n)[st(1^k)] (see A002872 for notation). - Andrey Zabolotskiy, Oct 17 2017 EXAMPLE Triangle begins:    1;    1,  1;    2,  3,  1;    5, 10,  6,  1;   15, 37, 31, 10,  1;   ... From Paul Barry, Jan 12 2009: (Start) Production array begins   1, 1;   1, 2, 1;   0, 2, 3, 1;   0, 0, 3, 4, 1;   0, 0, 0, 4, 5, 1;   ... (End) MAPLE a:= proc(n, k) option remember; `if`(k<0 or k>n, 0,       `if`(n=0, 1, a(n-1, k-1) +(k+1)*(a(n-1, k) +a(n-1, k+1))))     end: seq(seq(a(n, k), k=0..n), n=0..15);  # Alois P. Heinz, Nov 30 2012 MATHEMATICA a[n_, k_] = Sum[StirlingS2[n, i]*Binomial[i, k], {i, 0, n}]; Flatten[Table[a[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Aug 29 2011, after Vladeta Jovovic *) PROG (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)) (Sage) def A049020_triangle(dim):     M = matrix(ZZ, dim, dim)     for n in (0..dim-1): M[n, n] = 1     for n in (1..dim-1):         for k in (0..n-1):             M[n, k] = M[n-1, k-1]+(k+1)*M[n-1, k]+(k+1)*M[n-1, k+1]     return M A049020_triangle(9) # Peter Luschny, Sep 19 2012 CROSSREFS First column gives A000110, second column = A005493. Third column = A003128, row sums = A001861, A059340. See A244489 for another version of this triangle. Cf. A245109. Sequence in context: A060693 A172381 A089302 * A299105 A307899 A144634 Adjacent sequences:  A049017 A049018 A049019 * A049021 A049022 A049023 KEYWORD nonn,tabl,nice,easy,changed AUTHOR EXTENSIONS More terms from James A. Sellers. Better definition from Geoffrey Critzer, Nov 30 2012. STATUS approved

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Last modified April 20 06:31 EDT 2021. Contains 343121 sequences. (Running on oeis4.)