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A008299 Triangle T(n,k) of associated Stirling numbers of second kind, n>=2, 1<=k<=floor(n/2). 22
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 (list; graph; refs; listen; history; text; internal format)
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

Rows are of lengths 1,1,2,2,3,3,.... (This suggests that the diagonals are rows of a full lower triangular matrix. - 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.

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"

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

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

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms

T. Copeland, Short note on Lagrange inversion

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

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

MathStackExchange, Mahler polynomials and the zeros of the incomplete gamma function, a MathStackExchange 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).

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

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(S_r(n,k)*u^k*(t^n/n!), n>=0, k>=0)=exp(u*(e^t-sum(t^i/i!, i=0..r-1))).

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

EXAMPLE

There are 3 ways of partitioning a set N of cardinality 4 into 2 blocks each of cardinality at least 2, so S_2(4,2)=3. Table begins

.n\k.|..1....2.....3.....4

= = = = = = = = = = = = = =

..2..|..1

..3..|..1

..4..|..1....3

..5..|..1...10

..6..|..1...25....15

..7..|..1...56...105

..8..|..1..119...490...105

..9..|..1..246..1918..1260

...

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

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 *)

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.

Sequence in context: A211360 A178866 A019427 * A016478 A102430 A135573

Adjacent sequences:  A008296 A008297 A008298 * A008300 A008301 A008302

KEYWORD

nonn,tabf,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

More terms from David Wasserman, Jun 13 2007

Edited by Peter Bala, Dec 04 2011

STATUS

approved

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Last modified June 22 10:09 EDT 2017. Contains 288608 sequences.