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A008299 Triangle T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= k <= floor(n/2).

%I A008299 #165 Feb 16 2025 08:32:32
%S A008299 1,1,1,3,1,10,1,25,15,1,56,105,1,119,490,105,1,246,1918,1260,1,501,
%T A008299 6825,9450,945,1,1012,22935,56980,17325,1,2035,74316,302995,190575,
%U A008299 10395,1,4082,235092,1487200,1636635,270270,1,8177,731731,6914908,12122110
%N A008299 Triangle T(n,k) of associated Stirling numbers of second kind, n >= 2, 1 <= k <= floor(n/2).
%C A008299 T(n,k) is the number of set partitions of [n] into k blocks of size at least 2. Compare with A008277 (blocks of size at least 1) and A059022 (blocks of size at least 3). See also A200091. Reading the table by diagonals gives A134991. The row generating polynomials are the Mahler polynomials s_n(-x). See [Roman, 4.9]. - _Peter Bala_, Dec 04 2011
%C A008299 Row n gives coefficients of moments of Poisson distribution about the mean expressed as polynomials in lambda [Haight]. The coefficients of the moments about the origin are the Stirling numbers of the second kind, A008277. - _N. J. A. Sloane_, Jan 24 2020
%C A008299 Rows are of lengths 1,1,2,2,3,3,..., a pattern typical of matrices whose diagonals are rows of another lower triangular matrix--in this instance those of A134991. - _Tom Copeland_, May 01 2017
%C A008299 For a relation to decomposition of spin correlators see Table 2 of the Delfino and Vito paper. - _Tom Copeland_, Nov 11 2012
%D A008299 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
%D A008299 Frank Avery Haight, "Handbook of the Poisson distribution," John Wiley, 1967. See pages 6,7, but beware of errors. [Haight on page 7 gives five different ways to generate these numbers (see link)].
%D A008299 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
%D A008299 S. Roman, The Umbral Calculus, Dover Publications, New York (2005), pp. 129-130.
%H A008299 Vincenzo Librandi and Alois P. Heinz, <a href="/A008299/b008299.txt">Rows n = 2..200, flattened</a> (rows n = 2..104 from Vincenzo Librandi)
%H A008299 Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H A008299 Peter Bala, <a href="/A112007/a112007.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>
%H A008299 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 A008299 J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="https://doi.org/10.37236/4814">Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers</a>, Electronic Journal of Combinatorics 22(3) (2015), #P3.37.
%H A008299 Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Beyene/beyene13.html">Set Partitions and Other Bell Number Enumerated Objects</a>, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.
%H A008299 Gilles Bonnet and Anna Gusakova, <a href="https://arxiv.org/abs/2404.16756">Concentration inequalities for Poisson U-statistics</a>, arXiv:2404.16756 [math.PR], 2024. See p. 17.
%H A008299 E. Rodney Canfield, J. William Helton, and Jared A. Hughes, <a href="https://arxiv.org/abs/2409.01489">Uniform Convergence of an Asymptotic Approximation to Associated Stirling Numbers</a>, arXiv:2409.01489 [math.CO], 2024. See p. 12.
%H A008299 Tom Copeland, <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>
%H A008299 Tom Copeland, <a href="http://tcjpn.wordpress.com/2008/09/13/short-note-on-lagrange-inversion/">Short note on Lagrange inversion</a>
%H A008299 Robert Coquereaux and Jean-Bernard Zuber, <a href="https://arxiv.org/abs/2305.01100">Counting partitions by genus. II. A compendium of results</a>, arXiv:2305.01100 [math.CO], 2023. See p. 3.
%H A008299 Daniel W. Cranston and Chun-Hung Liu, <a href="https://arxiv.org/abs/2211.02818">Proper Conflict-free Coloring of Graphs with Large Maximum Degree</a>, arXiv:2211.02818 [math.CO], 2022.
%H A008299 Gesualdo Delfino and Jacopo Viti, <a href="http://arxiv.org/abs/1104.4323">Potts q-color field theory and scaling random cluster model</a>, arXiv:1104.4323 [hep-th], 2011.
%H A008299 Ming-Jian Ding and Jiang Zeng, <a href="https://arxiv.org/abs/2307.00566">Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions</a>, arXiv:2307.00566 [math.CO], 2023.
%H A008299 A. E. Fekete, <a href="http://www.jstor.org/stable/2974533">Apropos two notes on notation</a>, Amer. Math. Monthly, 101 (1994), 771-778.
%H A008299 F. A. Haight, <a href="/A008299/a008299_1.pdf">Handbook of the Poisson distribution</a>, John Wiley, 1967 [Annotated scan of page 7 only. Note that there is an error in the table.]
%H A008299 Mathematics Stack Exchange, <a href="http://math.stackexchange.com/questions/1597737/mahler-polynomials-and-zeros-of-the-incomplete-gamma-function">Mahler polynomials and the zeros of the incomplete gamma function</a>, a Mathematics Stack Exchange question by Tom Copeland, Jan 06 2016.
%H A008299 R. Paris, <a href="http://dx.doi.org/10.1016/S0377-0427(02)00553-8">A uniform asymptotic expansion for the incomplete gamma function</a>, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See 332. From Tom Copeland, Jan 03 2016).
%H A008299 Andrew Elvey Price and Alan D. Sokal, <a href="https://arxiv.org/abs/2001.01468">Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials</a>, arXiv:2001.01468 [math.CO], 2020.
%H A008299 E. G. Santos, <a href="https://arxiv.org/abs/2411.16492">Counting non-attacking chess pieces placements: Bishops and Anassas</a>, arXiv:2411.16492 [math.CO], 2024. See p. 2.
%H A008299 L. M. Smiley, <a href="http://arXiv.org/abs/math.CO/0006106">Completion of a Rational Function Sequence of Carlitz</a>, arXiv:math/0006106 [math.CO], 2000.
%H A008299 M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq. 14 (2011) # 11.9.7.
%H A008299 Erik Vigren and Andreas Dieckmann, <a href="https://doi.org/10.3390/sym14061090">A New Result in Form of Finite Triple Sums for a Series from Ramanujan’s Notebooks</a>, Symmetry (2022) Vol. 14, No. 6, 1090.
%H A008299 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MahlerPolynomial.html">Mahler polynomial</a>
%H A008299 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mahler_polynomials">Mahler polynomials</a>
%F A008299 T(n,k) = abs(A137375(n,k)).
%F A008299 E.g.f. with additional constant 1: exp(t*(exp(x)-1-x)) = 1 + t*x^2/2! + t*x^3/3! + (t+3*t^2)*x^4/4! + ....
%F A008299 Recurrence relation: T(n+1,k) = k*T(n,k) + n*T(n-1,k-1).
%F A008299 T(n,k) = A134991(n-k,k); A134991(n,k) = T(n+k,k).
%F A008299 More generally, if S_r(n,k) gives the number of set partitions of [n] into k blocks of size at least r then we have the recurrence S_r(n+1,k) = k*S_r(n,k) + binomial(n,r-1)*S_r(n-r+1,k-1) (for this sequence, r=2), with associated e.g.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*(t^n/n!) = exp(u*(e^t - Sum_{i=0..r-1} t^i/i!)).
%F A008299 T(n,k) = Sum_{i=0..k} (-1)^i*binomial(n, i)*Sum_{j=0..k-i} (-1)^j*(k-i-j)^(n-i)/(j!*(k-i-j)!). - _David Wasserman_, Jun 13 2007
%F A008299 G.f.: (R(0)-1)/(x^2*y), where R(k) = 1 - (k+1)*y*x^2/( (k+1)*y*x^2 - (1-k*x)*(1-x-k*x)/R(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 09 2013
%F A008299 T(n,k) = Sum_{i=0..min(n,k)} (-1)^i * binomial(n,i) * Stirling2(n-i,k-i) = Sum_{i=0..min(n,k)} (-1)^i * A007318(n,i) * A008277(n-i,k-i). - _Max Alekseyev_, Feb 27 2017
%F A008299 T(n, k) = Sum_{j=0..n-k} binomial(j, n-2*k)*E2(n-k, n-k-j) where E2(n, k) are the second-order Eulerian numbers A340556. - _Peter Luschny_, Feb 11 2021
%e A008299 There are 3 ways of partitioning a set N of cardinality 4 into 2 blocks each of cardinality at least 2, so T(4,2)=3.
%e A008299 Table begins:
%e A008299   1;
%e A008299   1;
%e A008299   1,    3;
%e A008299   1,   10;
%e A008299   1,   25,     15;
%e A008299   1,   56,    105;
%e A008299   1,  119,    490,     105;
%e A008299   1,  246,   1918,    1260;
%e A008299   1,  501,   6825,    9450,      945;
%e A008299   1, 1012,  22935,   56980,    17325;
%e A008299   1, 2035,  74316,  302995,   190575,   10395;
%e A008299   1, 4082, 235092, 1487200,  1636635,  270270;
%e A008299   1, 8177, 731731, 6914908, 12122110, 4099095, 135135;
%e A008299   ...
%e A008299 Reading the table by diagonals produces the triangle A134991.
%p A008299 A008299 := proc(n,k) local i,j,t1; if k<1 or k>floor(n/2) then t1 := 0; else
%p A008299 t1 := add( (-1)^i*binomial(n, i)*add( (-1)^j*(k - i - j)^(n - i)/(j!*(k - i - j)!), j = 0..k - i), i = 0..k); fi; t1; end; # _N. J. A. Sloane_, Dec 06 2016
%p A008299 G:= exp(lambda*(exp(x)-1-x)):
%p A008299 S:= series(G,x,21):
%p A008299 seq(seq(coeff(coeff(S,x,n)*n!,lambda,k),k=1..floor(n/2)),n=2..20); # _Robert Israel_, Jan 15 2020
%p A008299 T := proc(n, k) option remember; if n < 0 then return 0 fi; if k = 0 then return k^n fi; k*T(n-1, k) + (n-1)*T(n-2, k-1) end:
%p A008299 seq(seq(T(n,k), k=1..n/2), n=2..9); # _Peter Luschny_, Feb 11 2021
%t A008299 t[n_, k_] := Sum[ (-1)^i*Binomial[n, i]*Sum[ (-1)^j*(k - i - j)^(n - i)/(j!*(k - i - j)!), {j, 0, k - i}], {i, 0, k}]; Flatten[ Table[ t[n, k], {n, 2, 14}, {k, 1, Floor[n/2]}]] (* _Jean-François Alcover_, Oct 13 2011, after _David Wasserman_ *)
%t A008299 Table[Sum[Binomial[n, k - j] StirlingS2[n - k + j, j] (-1)^(j + k), {j, 0, k}], {n, 15}, {k, n/2}] // Flatten (* _Eric W. Weisstein_, Nov 13 2018 *)
%o A008299 (PARI) {T(n, k) = if( n < 1 || 2*k > n, n==0 && k==0, sum(i=0, k, (-1)^i * binomial( n, i) * sum(j=0, k-i, (-1)^j * (k-i-j)^(n-i) / (j! * (k-i-j)!))))}; /* _Michael Somos_, Oct 19 2014 */
%o A008299 (PARI) { T(n,k) = sum(i=0,min(n,k), (-1)^i * binomial(n,i) * stirling(n-i,k-i,2) ); } /* _Max Alekseyev_, Feb 27 2017 */
%Y A008299 Rows: A000247 (k=2), A000478 (k=3), A058844 (k=4).
%Y A008299 Row sums: A000296, diagonal: A259877.
%Y A008299 Cf. A059022, A059023, A059024, A059025, A134991, A137375, A200091, A269939, A340556.
%K A008299 nonn,tabf,nice,easy
%O A008299 2,4
%A A008299 _N. J. A. Sloane_
%E A008299 Formula and cross-references from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000
%E A008299 Edited by _Peter Bala_, Dec 04 2011
%E A008299 Edited by _N. J. A. Sloane_, Jan 24 2020