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

1, 1, 1, 3, 1, 10, 1, 25, 15, 1, 56, 105, 1, 119, 490, 105, 1, 246, 1918, 1260, 1, 501, 6825, 9450, 945, 1, 1012, 22935, 56980, 17325, 1, 2035, 74316, 302995, 190575, 10395, 1, 4082, 235092, 1487200, 1636635, 270270, 1, 8177, 731731, 6914908, 12122110

Offset

2,4

Comments

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

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

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

For a relation to decomposition of spin correlators see Table 2 of the Delfino and Vito paper. - Tom Copeland, Nov 11 2012

References

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.

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)].

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.

S. Roman, The Umbral Calculus, Dover Publications, New York (2005), pp. 129-130.

Links

Vincenzo Librandi and Alois P. Heinz, Rows n = 2..200, flattened (rows n = 2..104 from Vincenzo Librandi)

Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order"

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.

J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Generalized Stirling permutations and forests: Higher-order Eulerian and Ward numbers, Electronic Journal of Combinatorics 22(3) (2015), #P3.37.

Fufa Beyene, Jörgen Backelin, Roberto Mantaci, and Samuel A. Fufa, Set Partitions and Other Bell Number Enumerated Objects, J. Int. Seq., Vol. 26 (2023), Article 23.1.8.

Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms

Tom Copeland, Short note on Lagrange inversion

Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.

Daniel W. Cranston and Chun-Hung Liu, Proper Conflict-free Coloring of Graphs with Large Maximum Degree, arXiv:2211.02818 [math.CO], 2022.

Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv:1104.4323 [hep-th], 2011.

Ming-Jian Ding and Jiang Zeng, Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions, arXiv:2307.00566 [math.CO], 2023.

A. E. Fekete, Apropos two notes on notation, Amer. Math. Monthly, 101 (1994), 771-778.

F. A. Haight, Handbook of the Poisson distribution, John Wiley, 1967 [Annotated scan of page 7 only. Note that there is an error in the table.]

Mathematics Stack Exchange, Mahler polynomials and the zeros of the incomplete gamma function, a Mathematics Stack Exchange question by Tom Copeland, Jan 06 2016.

R. Paris, A uniform asymptotic expansion for the incomplete gamma function, Journal of Computational and Applied Mathematics, 148 (2002), p. 223-239 (See 332. From Tom Copeland, Jan 03 2016).

Andrew Elvey Price and Alan D. Sokal, Phylogenetic trees, augmented perfect matchings, and a Thron-type continued fraction (T-fraction) for the Ward polynomials, arXiv:2001.01468 [math.CO], 2020.

L. M. Smiley, Completion of a Rational Function Sequence of Carlitz, arXiv:math/0006106 [math.CO], 2000.

M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.

Erik Vigren and Andreas Dieckmann, A New Result in Form of Finite Triple Sums for a Series from Ramanujan’s Notebooks, Symmetry (2022) Vol. 14, No. 6, 1090.

Eric Weisstein's World of Mathematics, Mahler polynomial

Wikipedia, Mahler polynomials

Formula

T(n,k) = abs(A137375(n,k)).

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! + ....

Recurrence relation: T(n+1,k) = k*T(n,k) + n*T(n-1,k-1).

T(n,k) = A134991(n-k,k); A134991(n,k) = T(n+k,k).

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!)).

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

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

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

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

Example

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. Table begins:
1;
1;
1,    3;
1,   10;
1,   25,     15;
1,   56,    105;
1,  119,    490,     105;
1,  246,   1918,    1260;
1,  501,   6825,    9450,      945;
1, 1012,  22935,   56980,    17325;
1, 2035,  74316,  302995,   190575,   10395;
1, 4082, 235092, 1487200,  1636635,  270270;
1, 8177, 731731, 6914908, 12122110, 4099095, 135135;
...
Reading the table by diagonals produces the triangle A134991.

Maple

A008299 := proc(n, k) local i, j, t1; if k<1 or k>floor(n/2) then t1 := 0; else
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
G:= exp(lambda*(exp(x)-1-x)):
S:= series(G, x, 21):
seq(seq(coeff(coeff(S, x, n)*n!, lambda, k), k=1..floor(n/2)), n=2..20); # Robert Israel, Jan 15 2020
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:
seq(seq(T(n, k), k=1..n/2), n=2..9); # Peter Luschny, Feb 11 2021

Mathematica

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 *)
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 *)

Prog

(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 */
(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 */

Crossrefs

Rows: A000247 (k=2), A000478 (k=3), A058844 (k=4).

Row sums: A000296, diagonal: A259877.

Cf. A059022, A059023, A059024, A059025, A134991, A137375, A200091, A269939, A340556.

Sequence in context: A364850 A325830 A331155 * A016478 A135573 A257254

Adjacent sequences: A008296 A008297 A008298 * A008300 A008301 A008302

Keyword

nonn,tabf,nice,easy

Author

N. J. A. Sloane

Extensions

Formula and cross-references from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000

Edited by Peter Bala, Dec 04 2011

Edited by N. J. A. Sloane, Jan 24 2020

Status

approved