A061691 -id:A061691 - cp's OEIS frontend

A023998 Number of block permutations on an n-set which are uniform, i.e., corresponding blocks have same size.

1, 1, 3, 16, 131, 1496, 22482, 426833, 9934563, 277006192, 9085194458, 345322038293, 15024619744202, 740552967629021, 40984758230303149, 2527342803112928081, 172490568947825135203, 12952575262915522547136, 1064521056888312620947794, 95305764621957309071404877

Offset: 0

Des FitzGerald (D.FitzGerald(AT)utas.edu.au)

Number of games of simple patience with n cards. Take a shuffled deck of n cards labeled 1..n; as each card is dealt it is placed either on a higher-numbered card or starts a new pile to the right. Cards are not moved once they are placed. Suggested by reading Aldous and Diaconis. - N. J. A. Sloane, Dec 19 1999

Number of set partitions of [2n] such that within each block the numbers of odd and even elements are equal. a(2) = 3: 1234, 12|34, 14|23; a(3) = 16: 123456, 1234|56, 1236|45, 1245|36, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 1456|23, 14|2356, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34. - Alois P. Heinz, Jul 14 2016

			For n=3 there are 25 block permutations, of which 9 of the form ({1} maps to {1,2}; {2,3} maps to {3}), are not uniform. Hence a(3) = 25 - 9 = 16.
Alternatively, for n=3 the 6 permutations of 3 cards produce 16 games, as follows: 123 -> {1,2,3}; 132 -> {1,32}, {1,3,2}; 213 -> {21,3}, {2,1,3}; 231 -> {21,3}, {2,31}, {2,3,1}; 312 -> {31,2}, {32,1}, {3,1,2}; 321 -> {321}, {32,1}, {31,2}, {3,21}, {3,2,1}.
G.f.: A(x) = 1 + x + 3*x^2/2!^2 + 16*x^3/3!^2 + 131*x^4/4!^2 + 1496*x^5/5!^2 + ...
log(A(x)) = x + x^2/2!^2 + x^3/3!^2 + x^4/4!^2 + x^5/5!^2 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..300
M. Aguiar and R. C. Orellana, The Hopf algebra of uniform block permutations, J. Algebr. Comb. 28 (2008) 115-138
D. Aldous and P. Diaconis, Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc. 36 (1999), 413-432.
H. Cheballah, S. Giraudo, and R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
Fabian Faulstich, Bernd Sturmfels, and Svala Sverrisdóttir, Algebraic Varieties in Quantum Chemistry, arXiv:2308.05258 [math.AG], 2023.
D. G. FitzGerald and Jonathan Leech, Dual symmetric inverse monoids and representation theory, J. Australian Mathematical Society (Series A), Vol. 64 (1998), pp. 345-367.
Raúl E. González-Torres, A geometric study of cores of idempotent stochastic matrices, Linear Algebra Appl. 527, 87-127 (2017).
Rosa Orellana, Franco Saliola, Anne Schilling, and Mike Zabrocki, The lattice of submonoids of the uniform block permutations containing the symmetric group, arXiv:2405.09710 [math.CO], 2024. See p. 3.
Sebastian Volz, Design and Implementation of Efficient Algorithms for Operations on Partitions of Sets, Bachelor Thesis, Saarland Univ. (Germany, 2023). See p. 45.

Cf. A023997, A002720, A061691.
Cf. A132813.
Column k=2 of A275043.
Main diagonal of A321296 and of A322670.

a023998 n = a023998_list !! n
a023998_list = 1 : f 2 [1] a132813_tabl where
   f x ys (zs:zss) = y : f (x + 1) (ys ++ [y]) zss where
                     y = sum $ zipWith (*) ys zs
-- Reinhard Zumkeller, Apr 04 2014

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-i)*binomial(n-1, i-1)/i!, i=1..n))
    end:
a:= n-> b(n)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, May 11 2016

a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] Binomial[n-1, k] a[k], {k, 0, n-1}];
Array[a, 25, 0] (* Jean-François Alcover, Jul 28 2016 *)
nmax = 20; CoefficientList[Series[E^(-1 + BesselI[0, 2*Sqrt[x]]), {x, 0, nmax}], x]*Range[0, nmax]!^2 (* Vaclav Kotesovec, Jun 09 2019 *)

a(n)=if(n==0,1,sum(k=0,n-1,binomial(n,k)*binomial(n-1,k)*a(k))) \\ Paul D. Hanna, Aug 15 2007

{a(n)=n!^2*polcoeff(exp(sum(m=1, n, x^m/m!^2)+x*O(x^n)), n)} /* Paul D. Hanna */

N=66; x='x+O('x^N); /* that many terms */
Vec(serlaplace(serlaplace(exp(sum(n=1, N, x^n/n!^2))))) /* show terms */
/* Joerg Arndt, Jul 12 2011 */

v=vector(N); v[1]=1;
for (n=1,N-1, v[n+1]=sum(k=0,n-1, binomial(n,k)*binomial(n-1,k)*v[k+1]) );
v /* show terms */
/* Joerg Arndt, Jul 12 2011 */

a(n) = Sum_{k=0..n-1} C(n,k)*C(n-1,k)*a(k) for n>0 with a(0)=1. - Paul D. Hanna, Aug 15 2007
G.f.: Sum_{n>=0} a(n)*x^n/n!^2 = exp( Sum_{n>=1} x^n/n!^2 ). [Paul D. Hanna, Jan 04 2011; merged from duplicate entry A179119]
Row sums of A061691.
Generating function: Let J(z) = Sum_{n>=0} z^n/n!^2. Then exp(J(z)-1) = Sum_{n>=0} a(n)*z^n/n!^2 = 1 + z + 3*z^2/2!^2 + 16*z^3/3!^2 + .... - Peter Bala, Jul 11 2011

More terms from Vladeta Jovovic, Sep 03 2002

A192721 The number of pairs of permutations in the product group S_n X S_n with k common descents, n >= 1 and 0 <= k <= n-1.

Original entry on oeis.org

1, 3, 1, 19, 16, 1, 211, 299, 65, 1, 3651, 7346, 3156, 246, 1, 90921, 237517, 160322, 28722, 917, 1, 3081513, 9903776, 9302567, 2864912, 245407, 3424, 1, 136407699, 520507423, 632274183, 288196659, 46261609, 2041965, 12861, 1

Offset: 1

Peter Bala, Jul 11 2011

Let S_n denote the symmetric group on {1,2,...,n}. A permutation p_1p_2...p_n in S_n has a descent at position i (1 <= i <= n-1) if p_i > p_(i+1). The Eulerian numbers A008292 (with an offset of 0 in the column indexing) enumerate permutations by descents. We define a pair of permutations p_1p_2...p_n and q_1q_2...q_n to have a common descent at position i (1 <= i <= n-1) if both p_i > p_(i+1) and q_i > q_(i+1). For example, the permutations (3241) and (4231) in S_4 have common descents at positions i = 1 and i = 3. The table entry T(n,k) gives the number of pairs of permutations in the Cartesian product S_n x S_n with k common descents.

The generalized Stirling numbers associated with this triangle is A061691. See also A192722.

			The triangle begins
n/k|.....0.......1.......2......3....4.....5
============================================
..1|.....1
..2|.....3.......1
..3|....19......16.......1
..4|...211.....299......65......1
..5|..3651....7346....3156....246....1
..6|.90921..237517..160322..28722..917.....1
..
Row 3 entries T(3,0) = 19, T(3,1) = 16 and T(3,2) = 1 can be read from the following table:
============================================
Number of common descents in S_3 x S_3
============================================
.
...|.123...132...213...231...312...321
======================================
123|..0.....0.....0.....0.....0.....0
132|..0.....1.....0.....1.....0.....1
213|..0.....0.....1.....0.....1.....1
231|..0.....1.....0.....1.....0.....1
312|..0.....0.....1.....0.....1.....1
321|..0.....1.....1.....1.....1.....2
Matrix identity A192721 * A007318 = row reverse of A192722:
/...1................\ /..1..............\
|...3.....1...........||..1....1..........|
|..19....16.....1.....||..1....2....1.....|
|.211...299....65....1||..1....3....3....1|
|.....................||..................|
=
/...1...................\
|...4......1.............|
|..36.....18......1......|
|.576....432.....68.....1|
|........................|

Alois P. Heinz, Rows n = 1..45, flattened
L. Carlitz, R. Scoville and T. Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc., 80 (1974), 881-884.
L. Carlitz, R. Scoville, T. Vaughan, Enumeration of pairs of permutations, Discrete Math. 14, (1976) 215-239.
J-Marc Fedou and D. Rawlings, More statistics on permutation pairs, The Electronic Journal of Combinatorics, 1 (1994) #R11.
M. V. Koutras, Eulerian numbers associated with sequences of polynomials, Fibonacci Quart. 32 (1994) 44-57.
T. Mendes, J. Remmel, A. Riehl, A Generalization of the Generating Functions for Descent Statistic.
R. P. Stanley, Binomial posets, Möbius inversion and permutation enumeration, J. Combinat. Theory, A 20 (1976), 336-356.

Cf. A000275 (first column), A001044 (row sums), A008292, A008459, A061691, A192722.

#A192721
#J = sum {n>=0} z^n/n!^2
J := unapply(BesselJ(0, 2*I*sqrt(z)),z):
G := (1-x)/(-x + J(z*(x-1))):
Gser := simplify(series(G, z = 0, 12)):
for n from 1 to 10 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 10 do seq(coeff(P[n],x,k), k = 0..n-1) od;
# gives sequence in triangular form

max = 9; j[z_] := BesselJ[0, 2 I*Sqrt[z]]; g = (1 - x)/(-x + j[z*(x - 1)]); gser = Series[g, {z, 0, max}]; p[n_] := n!^2 Coefficient[ gser, z, n]; a[n_, k_] := Coefficient[ p[n], x, k]; Flatten[ Table[ a[n, k], {n, 1, max-1}, {k, 0, n-1}]] (* Jean-François Alcover, Dec 13 2011, after Maple *)

Generating function (Carlitz et al. 1976): Let J(z) = sum {n>=0} z^n/n!^2. Then (1-x)/(J(z*(x-1))-x) = 1 + sum {n>=1} (sum {k = 0..n-1} T(n,k)*x^k)*z^n/n!^2 = 1 + z + (3+x)*z^2/2!^2 + (19+16*x+x^2)*z^3/3!^2 + .... Define a polynomial sequence {p(n,x) }n>=0 by means of the generating function J(z)^x = sum {n>=0} p(n,x)*z^n/n!^2. The generalized Eulerian polynomials associated with the sequence {p(n,x)} as defined by [Koutras, 1994] are the polynomials sum {k = 0..n-1} T(n,k)*x^(n-k).
Relations with other sequences: The first column of the array (x*I-A008459)^-1 (I the identity matrix) is a sequence of rational functions whose numerator polynomials are the row generating polynomials for the present triangle. The change of variable x -> (x+1)/x followed by z -> x*z transforms the above bivariate generating function (1-x)/(J(z*(x-1))-x) into 1/(1+x-x*J(z)), which is the generating function for A192722. Equivalently, if we postmultiply the present triangle by Pascal's triangle A007318 we obtain the row reversed form of A192722: A192721 * A007318 = row reverse of A192722.
Row n sum = n!^2 = A001044(n).
First column [1,3,19,211,3651,...] = A000275 (apart from initial term).

A061692 Triangle of generalized Stirling numbers.

Original entry on oeis.org

1, 1, 4, 1, 27, 36, 1, 172, 864, 576, 1, 1125, 17500, 36000, 14400, 1, 7591, 351000, 1746000, 1944000, 518400, 1, 52479, 7197169, 80262000, 191394000, 133358400, 25401600, 1, 369580, 151633440, 3691514176, 17188416000, 23866214400, 11379916800, 1625702400

Offset: 1

N. J. A. Sloane, Jun 19 2001

			1; 1,4; 1,27,36; 1,172,864,576; ...

Alois P. Heinz, Rows n = 1..100, flattened
J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Diagonals give A001044, A061695, A061693, A061694. Cf. A061691.

Row sums give A061684.

b:= proc(n) option remember; expand(
      `if`(n=0, 1, add(x*b(n-i)/i!^3, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/i!, i=1..n))(b(n)*n!^3):
seq(T(n), n=1..10);  # Alois P. Heinz, Sep 10 2019

R[0, ] = 1; R[n, x_] := R[n, x] = x*Sum[Binomial[n, k]^2*Binomial[n-1, k]*R[k, x], {k, 0, n-1}]; Table[CoefficientList[R[n, x], x] // Rest, {n, 1, 8}] // Flatten (* Jean-François Alcover, Sep 01 2015, after Peter Bala *)

T(n, k) = 1/k!*Sum multinomial(n, n_1, n_2, ..n_k)^3, where the sum extends over all compositions (n_1, n_2, .., n_k) of n into exactly k nonnegative parts. - Vladeta Jovovic, Apr 23 2003

The row polynomials R(n,x) satisfy the recurrence equation R(n,x) = x*( sum {k = 0..n-1} binomial(n,k)^2*binomial(n-1,k)*R(k,x) ) with R(0,x) = 1. Also R(n,x + y) = sum {k = 0..n} binomial(n,k)^3*R(k,x)*R(n-k,y). - Peter Bala, Sep 17 2013

More terms from Vladeta Jovovic, Apr 23 2003

A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

Original entry on oeis.org

1, 1, 4, 1, 18, 36, 1, 68, 432, 576, 1, 250, 3900, 14400, 14400, 1, 922, 32400, 252000, 648000, 518400, 1, 3430, 262542, 3880800, 19404000, 38102400, 25401600, 1, 12868, 2119152, 56664384, 493920000, 1795046400, 2844979200, 1625702400

Offset: 1

Peter Bala, Jul 11 2011

Compare with triangle A019538, whose entries are given by
... Sum multinomial(n; n_1,n_2,...,n_k), where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.
For related tables see A061691 and A192721.
Let P be the poset of all ordered pairs (S,T) of subsets of [n] with |S|=|T|, ordered componentwise by inclusion.  T(n,k) is the number of length k chains in P from ({},{}) to ([n],[n]). - Geoffrey Critzer, Apr 15 2020

			The triangle begins
n/k|..1.....2.......3........4........5........6
================================================
.1.|..1
.2.|..1.....4
.3.|..1....18.....36
.4.|..1....68.....432......576
.5.|..1...250....3900....14400....14400
.6.|..1...922...32400...252000...648000...518400
...
T(4,2) = 68:
There are 3 compositions of 4 into 2 parts, namely, 4 = 2 + 2 = 1 + 3 = 3 + 1; hence
T(4,2) = (4!/(2!*2!))^2 + (4!/(1!*3!))^2 + (4!/(3!*1!))^2
= 36 + 16 + 16 = 68.
Matrix identity: A192721 * Pascal's triangle = row reverse of A192722:
/...1................\ /..1..............\
|...3.....1...........||..1....1..........|
|..19....16.....5.....||..1....2....1.....|
|.211...299....65....1||..1....3....3....1|
|.....................||..................|
=
/...1...................\
|...4......1.............|
|..36.....18......1......|
|.576....432.....68.....1|
|........................|

Alois P. Heinz, Rows n = 1..100, flattened

Cf. A001044, A002190, A061691, A192721, A102221 (row sums), A000275 (alternating row sums).

J := unapply(BesselJ(0, 2*sqrt(-1)*sqrt(z)), z):
G := 1/(1-x*(J(z)-1)):
Gser := simplify(series(G, z = 0, 15)):
for n from 1 to 14 do
P[n] := n!^2*sort(coeff(Gser, z, n)) od:
for n from 1 to 14 do seq(coeff(P[n], x, k), k = 1..n) od;
# yields sequence in triangular form
# second Maple program:
b:= proc(n) option remember; expand(
      `if`(n=0, 1, add(x*b(n-i)/i!^2, i=1..n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)*n!^2):
seq(T(n), n=1..14);  # Alois P. Heinz, Sep 10 2019

b[n_] := b[n] = Expand[If[n == 0, 1, Sum[x b[n-i]/i!^2, {i, 1, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][b[n] n!^2];
Table[T[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Generating function: Let J(z) = Sum_{n>=0} z^n/n!^2. Then
1 + Sum_{n>=1} (Sum_{k = 1..n} T(n,k)*x^k)*z^n/n!^2 = 1/(1 - x*(J(z) - 1))
  = 1 + x*z + (x + 4*x^2)*z^2/2!^2 + (x + 18*x^2 + 36*x^3)*z^3/3!^2 + ....
Relations with other sequences:
The change of variable z -> z/x followed by x -> 1/(x - 1) transforms the above bivariate generating function 1/(1 - x*(J(z) - 1)) into (1 - x)/(-x + J(z*(x-1))), which is the generating function for A192721.
1/k!*T(n,k) = A061691(n,k).
T(n,n) = n!^2 = A001044(n).
Row sums = A102221.
For n>=1, Sum_{k = 1..n} (-1)^(n+k)*T(n,k)/k = A002190(n).

A343254 Triangle read by rows: T(n,k) is the number of 2-balanced partitions of a set of n elements in which the first and the second subsets have cardinality k, for n >= 0, k = 0..floor(n/2).

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 15, 5, 3, 52, 15, 8, 203, 52, 25, 16, 877, 203, 89, 53, 4140, 877, 354, 197, 131, 21147, 4140, 1551, 810, 512, 115975, 21147, 7403, 3643, 2193, 1496, 678570, 115975, 38154, 17759, 10201, 6697, 4213597, 678570, 210803, 93130, 51146, 32345, 22482

Offset: 0

Francesca Aicardi, Jun 04 2021

A 2-balanced partition is a partition of a set which is the union of three subsets, with the property that the cardinality of the first two subsets are equal (possibly zero), and each block contains the same number (possibly zero) of elements from the first and from the second subset. The rows add to A344775.

T(n,0) are the Bell numbers. T(2k,k) are the numbers of 2-balanced partitions in the particular case in which the third set is empty. T(2k,k) are the generalized Bell numbers given in A023998.

			T(4,1) = 5, number of 2-balanced partitions of a set A of 4 elements with 1 element in the first subset and 1 element in the second subset: A={a} U {b} U {c,d}. The five partitions are: ((a,b),(c),(d)), ((a,b),(c,d)), ((a,b,c),(d)), ((a,b,d),(c)), ((a,b,c,d)). Note that if a block contains a, then it must contain b. Thus, T(n,1) = T(n-1,0).
Triangle T(n,k) begins:
        1;
        1;
        2,      1;
        5,      2;
       15,      5,      3;
       52,     15,      8;
      203,     52,     25,    16;
      877,    203,     89,    53;
     4140,    877,    354,   197,   131;
    21147,   4140,   1551,   810,   512;
   115975,  21147,   7403,  3643,  2193,  1496;
   678570, 115975,  38154, 17759, 10201,  6697;
  4213597, 678570, 210803, 93130, 51146, 32345, 22482;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
Francesca Aicardi, Balanced partitions, preprint on researchgate, 2021.
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Brauer and Jones tied monoids, arXiv:2107.04170 [math.RT], 2021.

Cf. A000110 (Bell numbers), A023998, A061691 (generalized Stirling numbers), A344775 (row sums).

T(n,k) = Sum_{j=1..n-k} C(n,k,j). C(n,k,j) is defined for n>=2k, j<=n-k, and obtained by the recursion: C(n,k,j) = C(n-1,k,j-1) + j*C(n-1,k,j), with initial conditions C(2k,k,j) = triangle A061691(k,j) of generalized Stirling numbers.

A061689 Generalized Stirling numbers.

Original entry on oeis.org

1, 34, 650, 10500, 161700, 2493120, 39372480, 644112000, 10977120000, 195392736000, 3636061228800, 70737918451200, 1437896940480000, 30512401920000000, 675213301002240000, 15564170478624768000, 373267990604500992000, 9302809053765427200000

Offset: 1

N. J. A. Sloane, Jun 18 2001

nonn

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Second subdiagonal of A061691.

More terms from Sean A. Irvine, Feb 27 2023

A061690 Generalized Stirling numbers.

Original entry on oeis.org

0, 0, 6, 72, 650, 5400, 43757, 353192, 2862330, 23352300, 191891117, 1587587760, 13215894133, 110619113423, 930376519256, 7858437064232, 66627124896218, 566791391339532, 4836144006188165, 41375938305568772, 354859541163656045

Offset: 1

N. J. A. Sloane, Jun 18 2001

nonn

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Third column of A061691.

a(n) = {n!^2*polcoef((besseli(0, 2*x + O(x^(2*n+1))) - 1)^3, 2*n)/6} \\ Andrew Howroyd, Mar 04 2021

a(n) = s(n, k) = Sum_{pi} n!/(k(1)! * 1!^k(1) * k(2)! * 2!^k(2) * ... * k(n)! * n!^k(n)) * (n!/(1!^k(1) * 2!^k(2) * ... * n!^k(n)))^L, where pi runs through all partitions k(1) +2 * k(2)+...+n * k(n) = n such that k = k(1)+k(2)+...+k(n), with k = 3 and L = 1.

Sum_{n>=1} a(n) * x^n / (n!)^2 = (BesselI(0,2*sqrt(x)) - 1)^3 / 6. - Ilya Gutkovskiy, Mar 04 2021

Formula and more terms from Vladeta Jovovic, Dec 09 2001

cp's OEIS Frontend

A023998 Number of block permutations on an n-set which are uniform, i.e., corresponding blocks have same size.

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A192721 The number of pairs of permutations in the product group S_n X S_n with k common descents, n >= 1 and 0 <= k <= n-1.

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A061692 Triangle of generalized Stirling numbers.

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A192722 T(n,k) = Sum of multinomial(n; n_1,n_2,...,n_k)^2, where the sum extends over all compositions (n_1,n_2,...,n_k) of n into exactly k nonnegative parts.

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A343254 Triangle read by rows: T(n,k) is the number of 2-balanced partitions of a set of n elements in which the first and the second subsets have cardinality k, for n >= 0, k = 0..floor(n/2).

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A061689 Generalized Stirling numbers.

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A061690 Generalized Stirling numbers.

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