A132393 -id:A132393 - cp's OEIS frontend

A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3x)^(-1/3), (-1/3)log(1 - 3*x)). A generalized Stirling1 triangle.

1, 1, 1, 4, 5, 1, 28, 39, 12, 1, 280, 418, 159, 22, 1, 3640, 5714, 2485, 445, 35, 1, 58240, 95064, 45474, 9605, 1005, 51, 1, 1106560, 1864456, 959070, 227969, 28700, 1974, 70, 1, 24344320, 42124592, 22963996, 5974388, 859369, 72128, 3514, 92, 1, 608608000, 1077459120, 616224492, 172323696, 27458613, 2662569, 159978, 5814, 117, 1, 17041024000, 30777463360, 18331744896, 5441287980, 941164860, 102010545, 7141953, 322770, 9090, 145, 1

Offset: 0

Wolfdieter Lang, May 18 2017

This is a generalization of the unsigned Stirling1 triangle A132393.
In general the lower triangular Sheffer matrix ((1 - d*x)^(-a/d), (-1/d)*log(1 - d*x)) is called here |S1hat[d,a]|. The signed matrix S1hat[d,a] with elements (-1)^(n-k)*|S1hat[d,a]|(n, k) is the inverse of the generalized Stirling2 Sheffer matrix S2hat[d,a] with elements S2[d,a](n, k)/d^k, where S2[d,a] is Sheffer (exp(a*x), exp(d*x) - 1).
In the Bala link the signed S1hat[d,a] (with row scaled elements S1[d,a](n,k)/d^n where S1[d,a] is the inverse matrix of S2[d,a]) is denoted by s_{(d,0,a)}, and there the notion exponential Riordan array is used for Sheffer array.
In the Luschny link the elements of |S1hat[m,m-1]| are called Stirling-Frobenius cycle numbers SF-C with parameter m.
From Wolfdieter Lang, Aug 09 2017: (Start)
The general row polynomials R(d,a;n,x) = Sum_{k=0..n} T(d,a;n,k)*x^k of the Sheffer triangle |S1hat[d,a]| satisfy, as special polynomials of the Boas-Buck class (see the reference), the identity (we use the notation of Rainville, Theorem 50, p. 141, adapted to an exponential generating function)
  (E_x - n*1)*R(d,a;n,x) = -n!*Sum_{k=0..n-1} d^k*(a*1 + d*beta(k)*E_x)*R(d,a;n-1-k,x)/(n-1-k)!, for n >= 0, with E_x = x*d/dx (Euler operator), and beta(k) = A002208(k+1)/A002209(k+1).
This entails a recurrence for the sequence of column k, for n > k >= 0: T(d,a;n,k) = (n!/(n - k))*Sum_{p=k..n-1} d^(n-1-p)*(a + d*k*beta(n-1-p))*T(d,a;p,k)/p!, with input T(d,a;k,k) = 1. For the present [d,a] = [3,1] case see the formula and example sections below. (End)
The inverse of the Sheffer triangular matrix S2[3,1] = A282629 is the Sheffer matrix S1[3,1] = (1/(1 + x)^(1/3), log(1 + x)/3) with rational elements S1[3,1](n, k) = (-1)^(n-m)*T(n, k)/3^n. - Wolfdieter Lang, Nov 15 2018

			The triangle T(n, k) begins:
n\k        0        1        2       3      4     5    6  7 8 ...
O:         1
1:         1        1
2:         4        5        1
3:        28       39       12       1
4:       280      418      159      22      1
5:      3640     5714     2485     445     35     1
6:     58240    95064    45474    9605   1005    51    1
7:   1106560  1864456   959070  227969  28700  1974   70  1
8:  24344320 42124592 22963996 5974388 859369 72128 3514 92 1
...
From _Wolfdieter Lang_, Aug 09 2017: (Start)
Recurrence: T(3, 1) = T(2, 0) + (3*3-2)*T(2, 1) = 4 + 7*5 = 39.
Boas-Buck recurrence for column k = 2 and n = 5:
T(5, 2) = (5!/3)*(3^2*(1 + 6*(3/8))*T(2,2)/2! + 3*(1 + 6*(5/12)*T(3, 2)/3! + (1 + 6*(1/2))* T(4, 2)/4!)) = (5!/3)*(9*(1 + 9/4)/2 + 3*(1 + 15/6)*12/6 + (1 + 3)*159/24) = 2485.
The beta sequence begins: {1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, ...}.
(End)

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.

P. Bala, A 3 parameter family of generalized Stirling numbers
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Peter Luschny, The Stirling-Frobenius numbers.

S2[d,a] for [d,a] = [1,0], [2,1], [3,1], [3,2], [4,1] and [4,3] is A048993, A154537, A282629, A225466, A285061 and A225467, respectively.
S2hat[d,a] for these [d,a] values is A048993, A039755, A111577 (offset 0), A225468, A111578 (offset 0) and A225469, respectively.
|S1hat[d,a]| for [d,a] = [1,0], [2,1], [3,2], [4,1] and [4,3] is A132393, A028338, A225470, A290317 and A225471, respectively.
Column sequences for k = 0..4: A007559, A024216(n-1), A286721(n-2), A382984, A382985.
Diagonal sequences: A000012, A000326(n+1), A024212(n+1), A024213(n+1).
Row sums: A008544. Alternating row sums: A000007.
Beta sequence: A002208(n+1)/A002209(n+1).

T[n_ /; n >= 1, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (3*n-2)* T[n-1, k]; T[, -1] = 0; T[0, 0] = 1; T[n, k_] /; nJean-François Alcover, Jun 20 2018 *)

Recurrence: T(n, k) = T(n-1, k-1) + (3*n-2)*T(n-1, k), for n >= 1, k = 0..n, and T(n, -1) = 0, T(0, 0) = 1 and T(n, k) = 0 for n < k.
E.g.f. of row polynomials R(n, x) = Sum_{k=0..n} T(n, k)*x^k (i.e., e.g.f. of the triangle) is (1 - 3*z)^{-(x+1)/3}.
E.g.f. of column k is (1 - 3*x)^(-1/3)*((-1/3)*log(1 - 3*x))^k/k!.
Recurrence for row polynomials is R(n, x) = (x+1)*R(n-1, x+3), with R(0, x) = 1.
Row polynomial R(n, x) = risefac(3,1;x,n) with the rising factorial
risefac(d,a;x,n) := Product_{j=0..n-1} (x + (a + j*d)). (For the signed case see the Bala link, eq. (16)).
T(n, k) = sigma^{(n)}{n-k}(a_0,a_1,...,a{n-1}) with the elementary symmetric functions with indeterminates a_j = 1 + 3*j.
T(n, k) = Sum_{j=0..n-k} binomial(n-j, k)*|S1|(n, n-j)*3^j, with the unsigned Stirling1 triangle |S1| = A132393.
Boas-Buck column recurrence (see a comment above): T(n, k) =
(n!/(n - k))*Sum_{p=k..n-1} 3^(n-1-p)*(1 + 3*k*beta(n-1-p))*T(p, k)/p!, for n > k >= 0, with input T(k, k) = 1, with beta(k) = A002208(k+1)/A002209(k+1). See an example below. - Wolfdieter Lang, Aug 09 2017

A052801 A simple grammar: labeled pairs of sequences of cycles.

Original entry on oeis.org

1, 2, 8, 46, 342, 3108, 33324, 411360, 5741856, 89379120, 1534623936, 28804923024, 586686138384, 12885385945248, 303537419684064, 7633673997722496, 204125888803996800, 5782960189212871680

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Robert Israel, Table of n, a(n) for n = 0..417
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 759.

Cf. A215916.

Cf. A007840, A354122, A354123.

spec := [S,{C=Cycle(Z),B=Sequence(C),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

CoefficientList[Series[1/(1+Log[1-x])^2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2013 *)

makelist(sum((-1)^(n-k)*stirling1(n, k)*(k+1)!, k, 0, n), n, 0, 17); /* Bruno Berselli, May 25 2011 */

E.g.f.: 1/(-1+log(-1/(-1+x)))^2.
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling1(n, k)*(k+1)!. - Vladeta Jovovic, Sep 21 2003
a(n) = D^n(1/(1-x)^2) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A052811. - Peter Bala, Nov 25 2011
a(n) ~ n! * n*exp(n)/(exp(1)-1)^(n+2). - Vaclav Kotesovec, Sep 30 2013
From Anton Zakharov, Aug 07 2016: (Start)
a(n) = A007840(n) + A215916(n).
a(n) = Sum_{k=2..n+1} k!*s(n,k) where s(n,k) is the unsigned Stirling number of the first kind, (A132393). (End)
a(0) = 1; a(n) = Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Nov 19 2023

A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 26, 36, 12, 1, 0, 150, 250, 120, 20, 1, 0, 1082, 2040, 1230, 300, 30, 1, 0, 9366, 19334, 13650, 4270, 630, 42, 1, 0, 94586, 209580, 166376, 62160, 11900, 1176, 56, 1, 0, 1091670, 2562354, 2229444, 952728, 220500, 28476, 2016, 72, 1

Offset: 0

Wolfdieter Lang, May 04 2007

Matrix product of Stirling2 with unsigned Stirling1 triangle.
For the subtriangle without column no. m=0 and row no. n=0 see A079641.
The reversed matrix product |S1|. S2 is given in A111596.
As a product of lower triangular Jabotinsky matrices this is a lower triangular Jabotinsky matrix. See the D. E. Knuth references given in A039692 for Jabotinsky type matrices.
E.g.f. for row polynomials P(n,x):=sum(a(n,m)*x^m,m=0..n) is 1/(2-exp(z))^x. See the e.g.f. for the columns given below.
A048993*A132393 as infinite lower triangular matrices. - Philippe Deléham, Nov 01 2009
Triangle T(n,k), read by rows, given by (0,2,1,4,2,6,3,8,4,10,5,...) DELTA (1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,...) where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 19 2011.
Also the Bell transform of A000629. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Triangle begins:
  1;
  0,    1;
  0,    2,    1;
  0,    6,    6,    1;
  0,   26,   36,   12,   1;
  0,  150,  250,  120,  20,  1;
  0, 1082, 2040, 1230, 300, 30,  1;

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened).
Olivier Bodini, Antoine Genitrini, Cécile Mailler, Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First ten rows and more.

Columns k=0..3 give A000007, A000629(n-1), A129063, A129064.

Cf. A000629, A000670, A005649, A079641, A129062, A325872, A325873, A383140.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> polylog(-n,1/2), 9); # Peter Luschny, Jan 27 2016

rows = 9;
t = Table[PolyLog[-n, 1/2], {n, 0, rows}]; T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)
p[n_] := Sum[StirlingS2[n, k] Pochhammer[x, k], {k, 0, n}];
Table[CoefficientList[FunctionExpand[p[n]], x], {n, 0, 9}] // Flatten (* Peter Luschny, Jun 27 2019 *)

def a_row(n):
    s = sum(stirling_number2(n,k)*rising_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

a(n,m) = Sum_{k=m..n} S2(n,k) * |S1(k,m)|, n>=0; S2=A048993, S1=A048994.
E.g.f. of column k (with leading zeros): (f(x)^k)/k! with f(x):= -log(1-(exp(x)-1)) = -log(2-exp(x)).
Sum_{0<=k<=n} T(n,k)*x^k = A153881(n+1), A000007(n), A000670(n), A005649(n) for x = -1,0,1,2 respectively. - Philippe Deléham, Nov 19 2011

New name by Peter Luschny, Jun 27 2019

A144739 7-factorial numbers A114799(7n+3): Partial products of A017017(k) = 7k+3, a(0) = 1.

Original entry on oeis.org

1, 3, 30, 510, 12240, 379440, 14418720, 648842400, 33739804800, 1990648483200, 131382799891200, 9590944392057600, 767275551364608000, 66752972968720896000, 6274779459059764224000, 633752725365036186624000, 68445294339423908155392000, 7871208849033749437870080000

Offset: 0

Philippe Deléham, Sep 20 2008

nonn

			a(0)=1, a(1)=3, a(2)=3*10=30, a(3)=3*10*17=510, a(4)=3*10*17*24=12240, ...

G. C. Greubel, Table of n, a(n) for n = 0..335
Index entries for sequences related to factorial numbers.

Cf. A114799, A001710, A001147, A032031, A008545, A047056, A011781, A045754, A084947, A144827, A147585, A049209, A051188.

List([0..20], n-> Product([0..n-1], k-> 7*k+3) ); # G. C. Greubel, Aug 19 2019

[ 1 ] cat [ &*[ (7*k+3): k in [0..n] ]: n in [0..20] ]; // Klaus Brockhaus, Nov 10 2008

a:= n-> product(7*j+3, j=0..n-1); seq(a(n), n=0..20); # G. C. Greubel, Aug 19 2019

Table[7^n*Pochhammer[3/7, n], {n,0,20}] (* G. C. Greubel, Aug 19 2019 *)

a(n)=prod(i=1,n,7*i-4) \\ Charles R Greathouse IV, Jul 02 2013

[product(7*k+3 for k in (0..n-1)) for n in (0..20)] # G. C. Greubel, Aug 19 2019

a(n) = Sum_{k=0..n} A132393(n,k)*3^k*7^(n-k).
G.f.: 1/(1-3*x/(1-7*x/(1-10*x/(1-14*x/(1-17*x/(1-21*x/(1-24*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-4)^n*Sum_{k=0..n} (7/4)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(3/7).
a(n) ~ sqrt(2*Pi)*7^n*n^n/(exp(n)*n^(1/14)*Gamma(3/7)). (End)
a(n) = A114799(7*n-4). - M. F. Hasler, Feb 23 2018
D-finite with recurrence: a(n) +(-7*n+4)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
Sum_{n>=0} 1/a(n) = 1 + (e/7^4)^(1/7)*(Gamma(3/7) - Gamma(3/7, 1/7)). - Amiram Eldar, Dec 19 2022

A164652 Triangle read by rows: Hultman numbers: a(n,k) is the number of permutations of n elements whose cycle graph (as defined by Bafna and Pevzner) contains k cycles for n >= 0 and 1 <= k <= n+1.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 8, 0, 15, 0, 1, 0, 84, 0, 35, 0, 1, 180, 0, 469, 0, 70, 0, 1, 0, 3044, 0, 1869, 0, 126, 0, 1, 8064, 0, 26060, 0, 5985, 0, 210, 0, 1, 0, 193248, 0, 152900, 0, 16401, 0, 330, 0, 1, 604800, 0, 2286636, 0, 696905, 0, 39963, 0, 495, 0, 1, 0, 19056960, 0, 18128396, 0, 2641925, 0, 88803, 0, 715, 0, 1

Offset: 0

Anthony Labarre, Aug 19 2009

a(n,k) is also the number of ways to express a given (n+1)-cycle as the product of an (n+1)-cycle and a permutation with k cycles (see Doignon and Labarre). a(n,n+1-2k) is the number of permutations of n elements whose block-interchange distance is k (see Christie, Doignon and Labarre).
Named after the Swedish mathematician Axel Hultman. - Amiram Eldar, Jun 11 2021
a(2*n,1) is the number of spanning trees in certain graphs with 2*n+1 vertices and n*(n+1) edges (see Ishikawa, Miezaki, and Tanaka). - Tsuyoshi Miezaki, Feb 08 2023

			Triangle begins:
  n=0:  1;
  n=1:  0, 1;
  n=2:  1, 0, 1;
  n=3:  0, 5, 0, 1;
  n=4:  8, 0, 15, 0, 1;
  n=5:  0, 84, 0, 35, 0, 1;
  n=6:  180, 0, 469, 0, 70, 0, 1;
  n=7:  0, 3044, 0, 1869, 0, 126, 0, 1;
  n=8:  8064, 0, 26060, 0, 5985, 0, 210, 0, 1;
  n=9:  0, 193248, 0, 152900, 0, 16401, 0, 330, 0, 1;
  n=10: 604800, 0, 2286636, 0, 696905, 0, 39963, 0, 495, 0, 1;
  ...
From _Jon E. Schoenfield_, May 20 2023: (Start)
As a right-aligned triangle:
                                                      1; n=0
                                                   0, 1; n=1
                                                1, 0, 1; n=2
                                           0,   5, 0, 1; n=3
                                        8, 0,  15, 0, 1; n=4
                                 0,    84, 0,  35, 0, 1; n=5
                            180, 0,   469, 0,  70, 0, 1; n=6
                      0,   3044, 0,  1869, 0, 126, 0, 1; n=7
                8064, 0,  26060, 0,  5985, 0, 210, 0, 1; n=8
          0,  193248, 0, 152900, 0, 16401, 0, 330, 0, 1; n=9
  604800, 0, 2286636, 0, 696905, 0, 39963, 0, 495, 0, 1; n=10
  ...
(End)

Axel Hultman, Toric permutations, Master's thesis, Department of Mathematics, KTH, Stockholm, Sweden, 1999.

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
Nikita Alexeev, Anna Pologova, and Max A. Alekseyev, Generalized Hultman Numbers and Cycle Structures of Breakpoint Graphs, Journal of Computational Biology, Vol. 24, No. 2 (2017), pp. 93-105; arXiv preprint, arXiv:1503.05285 [q-bio.GN], 2015-2017.
Nikita Alexeev and Peter Zograf, Hultman numbers, polygon gluings and matrix integrals, arXiv preprint arXiv:1111.3061 [math.PR], 2011.
Nikita Alexeev and Peter Zograf, Random matrix approach to the distribution of genomic distance, Journal of Computational Biology, Vol. 21, No. 8 (2014), pp. 622-631.
Miklos Bona and Ryan Flynn, The Average Number of Block Interchanges Needed to Sort A Permutation and a recent result of Stanley, arXiv:0811.0740 [math.CO], 2008.
Miklos Bona and Ryan Flynn, The Average Number of Block Interchanges Needed to Sort A Permutation and a recent result of Stanley, Inf. Process. Lett., Vol. 109 (2009), pp. 927-931.
David A. Christie, Sorting Permutations by Block-Interchanges, Inf. Process. Lett., Vol. 60, No. 4 (1996), pp. 165-169.
Robert Cori and Gábor Hetyei, On reduced unicellular hypermonopoles, arXiv:2403.19569 [math.CO], 2024. See at p. 4.
Jean-Paul Doignon and Anthony Labarre, On Hultman Numbers, J. Integer Seq., Vol. 10 (2007), Article 07.6.2, 13 pages.
Simona Grusea and Anthony Labarre, The distribution of cycles in breakpoint graphs of signed permutations, arXiv:1104.3353 [cs.DM], 2011-2012.
Reina Ishikawa, Tsuyoshi Miezaki, and Yuuho Tanaka, A new interpretation of Hultman numbers.

Cf. A000142 (row sums), A000217, A060593, A128174, A130534, A132393, A185259, A189507, A260695.

Cf. A185263 (rows reversed without 0's).

a164652 n k = a164652_tabl !! n !! k
a164652_row n = a164652_tabl !! n
a164652_tabl = [0] : tail (zipWith (zipWith (*)) a128174_tabl $
   zipWith (map . flip div) (tail a000217_list) (map init $ tail a130534_tabl))
-- Reinhard Zumkeller, Aug 01 2014

A164652:= (n, k)-> `if`(n-k mod 2 = 1, -Stirling1(n+2, k)/binomial(n+2, 2), 0):
for n from 0 to 7 do seq(A164652(n,k),k=1..n+1) od; # Peter Luschny, Mar 22 2015

T[n_, k_] := If[OddQ[n-k], Abs[StirlingS1[n+2, k]]/Binomial[n+2, 2], 0];
Table[T[n, k], {n, 0, 11}, {k, 1, n+1}] // Flatten (* Jean-François Alcover, Aug 10 2018 *)

T(n,k)= my(s=(n-k)%2); (-1)^s*s*stirling(n+2,k,1)/binomial(n+2,2);
concat(vector(12, n, vector(n, k, T(n-1,k)))) \\ Gheorghe Coserea, Jan 23 2018

def A164652(n, k):
    return stirling_number1(n+2,k)/binomial(n+2,2) if is_odd(n-k) else 0
for n in (0..7): print([A164652(n,k) for k in (1..n+1)]) # Peter Luschny, Mar 22 2015

T(n,k) = S1(n+2,k)/C(n+2,2) if n-k is odd, and 0 otherwise. Here S1(n,k) are the unsigned Stirling numbers of the first kind A132393 and C(n,k) is the binomial coefficient (see Bona and Flynn).
For n > 0: T(n,k) = A128174(n+1,k) * A130534(n+1,k-1) / A000217(n+1). - Reinhard Zumkeller, Aug 01 2014
n-th row polynomial R(n,x) = (x/2)*( P(n+1,x) + (-1)^n * P(n+1,-x) ) / binomial(n+2,2), where P(k,x) = (x + 1)*(x + 2)*...*(x + k). - Peter Bala, May 14 2023

T(0,1) set to 1 by Peter Luschny, Mar 24 2015

Edited to match values of k to the range 1 to n+1. - Max Alekseyev, Nov 20 2020

A285849 Number T(n,k) of permutations of [n] with k ordered cycles such that equal-sized cycles are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 6, 1, 0, 6, 19, 18, 1, 0, 24, 100, 105, 40, 1, 0, 120, 508, 1005, 430, 75, 1, 0, 720, 3528, 6762, 6300, 1400, 126, 1, 0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1, 0, 40320, 219168, 558548, 706608, 431445, 105336, 9114, 288, 1

Offset: 0

Alois P. Heinz, Apr 27 2017

Each cycle is written with the smallest element first and equal-sized cycles are arranged in increasing order of their first elements.

			T(3,1) = 2: (123), (132).
T(3,2) = 6: (1)(23), (23)(1), (2)(13), (13)(2), (3)(12), (12)(3).
T(3,3) = 1: (1)(2)(3).
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    1,     1;
  0,    2,     6,     1;
  0,    6,    19,    18,     1;
  0,   24,   100,   105,    40,     1;
  0,  120,   508,  1005,   430,    75,    1;
  0,  720,  3528,  6762,  6300,  1400,  126,   1;
  0, 5040, 24876, 61572, 62601, 28700, 3822, 196, 1;

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-10 give: A000007, A104150, A285853, A285854, A285855, A285856, A285857, A285858, A285859, A285860, A285861.
Row sums give A196301.
Main diagonal and first lower diagonal give: A000012, A002411.
T(2n,n) gives A285862.
Cf. A132393, A285824.

b:= proc(n, i, p) option remember; expand(`if`(n=0 or i=1,
      (p+n)!/n!*x^n, add(b(n-i*j, i-1, p+j)*(i-1)!^j*combinat
      [multinomial](n, n-i*j, i$j)/j!^2*x^j, j=0..n/i)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
seq(T(n), n=0..12);

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_, p_] := b[n, i, p] = Expand[If[n == 0 || i == 1, (p + n)!/n!*x^n, Sum[b[n - i*j, i - 1, p + j]*(i - 1)!^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!^2*x^j, {j, 0, n/i}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, n, 0]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 28 2018, after Alois P. Heinz *)

A342111 a(n) = (-1)^n * Sum_{k=0..n} Stirling1(n,k) * Stirling1(n,n-k).

Original entry on oeis.org

1, 0, 1, 12, 193, 3980, 100805, 3034920, 105994833, 4215106728, 188097696345, 9309515255700, 506149663220641, 29989851619249236, 1923467938147053389, 132771455705186298000, 9814431285244231295265, 773520674985391641371280, 64752473306596841023424945

Offset: 0

Vaclav Kotesovec, Feb 28 2021

nonn

G. C. Greubel, Table of n, a(n) for n = 0..350

Cf. A008275, A014322, A047796, A132393, A342110.
Cf. A048994, A155826.
Cf. A129256, A187235, A246117.

[(&+[(-1)^n*StirlingFirst(n, k)*StirlingFirst(n, n-k): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 03 2021

Table[(-1)^n*Sum[StirlingS1[n, k]*StirlingS1[n, n-k], {k, 0, n}], {n, 0, 20}]

a(n) = (-1)^n*sum(k=0, n, stirling(n, k, 1)*stirling(n, n-k, 1)); \\ Michel Marcus, Feb 28 2021

[sum( stirling_number1(n, k)*stirling_number1(n, n-k) for k in (0..n) ) for n in (0..30)] # G. C. Greubel, Jun 03 2021

a(n) ~ c * d^n * (n-1)!, where
d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.9108149645682558987515348052403521978987052817678471761394112...
c = 1/(4*sqrt(-LambertW(-1, -exp(-1/2)/2)) * sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.06903826111269387517867145566264007373042059749428879149076344304196548... - Vaclav Kotesovec, Feb 28 2021, updated May 14 2025
a(n) = [x^n] Product_{k=0..n-1} (1 + k*x)^2. - Seiichi Manyama, May 13 2025

A067318 Sum of the reflection lengths of all permutations of n letters.

Original entry on oeis.org

0, 1, 7, 46, 326, 2556, 22212, 212976, 2239344, 25659360, 318540960, 4261576320, 61148511360, 937030429440, 15275952518400, 264030355814400, 4823280687052800, 92865738644582400, 1879691760950169600, 39905092126771200000, 886664974825728000000

Offset: 1

H. Nick Hann (nickhann(AT)aol.com), Jan 15 2002

The reflection length of a permutation is the minimum number of transpositions needed to express the permutation.
May also be called the "weight" of the symmetric group S_n.
a(n) is the number of n+1-permutations that have at least 3 cycles. a(n) = Sum_{k=3..n+1} A132393(n+1,k). Cf. A001563 (n-permutations with at least 2 cycles). - Geoffrey Critzer, Dec 01 2013
The number of covering relations in the middle order on S_n. - Bridget Tenner, Jul 11 2025

			a(3)=7 since the permutations are 1, (12), (13), (23), (12)(13) and (13)(12). The sum of reflection lengths of all elements in S_3 is 0+1+1+1+2+2=7.
The terms satisfy the series: x/(1-x) = x/((1+x)*(1+2*x)*(1+3*x)) + 7*x^2/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)) + 46*x^3/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)) + 326*x^4/((1+x)*(1+2*x)*(1+3*x)*(1+4*x)*(1+5*x)*(1+6*x)) + ... - _Paul D. Hanna_, Aug 28 2012

N. Hann, Average Weight of a Random Permutation, preprint, 2002. [Apparently unpublished]

G. C. Greubel, Table of n, a(n) for n = 1..445 (terms 1..100 from T. D. Noe).
Mathilde Bouvel, Luca Ferrari, and Bridget Eileen Tenner, Between weak and Bruhat: the middle order on permutations, Graphs and Combin., 41 (2025), #34.
Emma Colaric, Ryan DeMuse, Jeremy L. Martin, and Mei Yin, Interval parking functions, arXiv:2006.09321 [math.CO], 2020.
Walter Feit, Roger Lyndon, and Leonard L. Scott, A remark on permutations, Journal of Combinatorial Theory (A) 18 234-235 (1975).
Loïc Foissy, The antipode of of [sic] a Com-PreLie Hopf algebra, arXiv:2406.01120 [math.CO], 2024. See p. 9.
Briana Foster-Greenwood and Cathy Kriloff, Spectra of Cayley Graphs of Complex Reflection Groups, arXiv preprint arXiv:1502.07392 [math.CO], 2015. (See remarks following Cor. 4.6.)
H. N. Hann, Symmetric Canonical Form [broken link]
Roberto Mantaci and Fanja Rakotondrajao, A permutation representation that knows what "Eulerian" means, Discrete Mathematics and Theoretical Computer Science, 4 101-108, (2001).

Cf. A000254, A001563, A001705, A062119, A067369, A067370, A084938.

ZL :=[S, {S = Set(Cycle(Z),3 <= card)}, labelled]: seq(combstruct[count](ZL, size=n), n=2..22); # Zerinvary Lajos, Mar 25 2008

a[n_] := n!*(n - HarmonicNumber[n]); Table[a[n], {n, 1, 21}](* Jean-François Alcover, Feb 10 2012 *)
nn=22;Drop[Range[0,nn]!CoefficientList[Series[1/(1-x)-1-Log[1/(1-x)]-Log[1/(1-x)]^2/2!,{x,0,nn}],x],2] (* Geoffrey Critzer, Dec 01 2013 *)

A067318(n):=n*n! - abs(stirling1(n+1, 2))$
makelist(A067318(n),n,1,30); /* Martin Ettl, Nov 03 2012 */

{a(n)=if(n==0,0,if(n==1, 1, 1-polcoeff(sum(k=1, n-1, a(k)*x^k/prod(j=1, k+2, (1+j*x+x*O(x^n)) ) ), n)))} /* Paul D. Hanna, Aug 28 2012 */

a(n) = n!*(0/1+1/2+...+(n-1)/n) = n!*(n - H_n), where H_n = Sum_{k=1..n} 1/k; a(1) = 0, a(2) = 1, a(n) = n*a(n-1) + (n-1)*(n-1)!.
a(n) = n*n! - abs(stirling1(n+1, 2)) (cf. A000254). E.g.f.: (x+(1-x)*log(1-x))/(1-x)^2. - Vladeta Jovovic, Feb 01 2003
a(n) = T(n, n-1) for the triangle read by rows: [0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Nov 30 2003
G.f.: x/(1-x) = Sum_{n>=1} a(n)*x^n/Product_{k=1..n+2} (1+k*x). - Paul D. Hanna, Aug 28 2012
a(n) = A062119(n) - A001705(n-1). - Anton Zakharov, Sep 22 2016

Definition and example edited by Bridget Tenner, Jul 11 2025

A144827 Partial products of successive terms of A017029; a(0)=1.

Original entry on oeis.org

1, 4, 44, 792, 19800, 633600, 24710400, 1136678400, 60243955200, 3614637312000, 242180699904000, 17921371792896000, 1451631115224576000, 127743538139762688000, 12135636123277455360000, 1237834884574300446720000, 134924002418598748692480000, 15651184280557454848327680000

Offset: 0

Philippe Deléham, Sep 21 2008

nonn

			a(0)=1, a(1)=4, a(2)=4*11=44, a(3)=4*11*18=792, a(4)=4*11*18*25=19800, ...

T. D. Noe, Table of n, a(n) for n = 0..100

Cf. A001715, A002866, A007559, A008546, A045754, A047053, A132393.

Cf. A049209, A049308, A051188, A084947, A144739, A147585.

[ 1 ] cat [ &*[ (7*k+4): k in [0..n] ]: n in [0..14] ]; // Klaus Brockhaus, Nov 10 2008

FoldList[Times,1,Range[4,150,7]] (* Harvey P. Dale, Apr 25 2014 *)

[1]+[4*7^(n-1)*rising_factorial(11/7, n-1) for n in (1..30)] # G. C. Greubel, Feb 22 2022

a(n) = Sum_{k=0..n} A132393(n,k)*4^k*7^(n-k).
G.f.: 1/(1-4*x/(1-7*x/(1-11*x/(1-14*x/(1-18*x/(1-21*x/(1-25*x/(1-... (continued fraction). - Philippe Deléham, Jan 08 2012
a(n) = (-3)^n*Sum_{k=0..n} (7/3)^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
From Ilya Gutkovskiy, Mar 23 2017: (Start)
E.g.f.: 1/(1 - 7*x)^(4/7).
a(n) ~ sqrt(2*Pi)*7^n*n^(n+1/14)/(exp(n)*Gamma(4/7)). (End)
a(n) = 4*7^(n-1)*Pochhammer(n-1, 11/7) with a(0) = 1. - G. C. Greubel, Feb 22 2022
Sum_{n>=0} 1/a(n) = 1 + (e/7^3)^(1/7)*(Gamma(4/7) - Gamma(4/7, 1/7)). - Amiram Eldar, Dec 19 2022

Corrected a(9) by Vincenzo Librandi, Jul 14 2011

A344855 Number T(n,k) of permutations of [n] having k cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 4, 11, 6, 1, 0, 8, 40, 35, 10, 1, 0, 16, 148, 195, 85, 15, 1, 0, 32, 560, 1078, 665, 175, 21, 1, 0, 64, 2160, 5992, 5033, 1820, 322, 28, 1, 0, 128, 8448, 33632, 37632, 17913, 4284, 546, 36, 1, 0, 256, 33344, 190800, 280760, 171465, 52941, 9030, 870, 45, 1

Offset: 0

Alois P. Heinz, May 30 2021

The sequence of column k satisfies a linear recurrence with constant coefficients of order k*(k+1)/2 = A000217(k).

			T(4,1) = 4: (1234), (1243), (1423), (1432).
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,    1;
  0,  2,    3,    1;
  0,  4,   11,    6,    1;
  0,  8,   40,   35,   10,    1;
  0, 16,  148,  195,   85,   15,   1;
  0, 32,  560, 1078,  665,  175,  21,  1;
  0, 64, 2160, 5992, 5033, 1820, 322, 28, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-10 give: A000007, A166444, A346317, A346318, A346319, A346320, A346321, A346322, A346323, A346324, A346325.
Row sums give A187251.
Main diagonal gives A000012, lower diagonal gives A000217, second lower diagonal gives A000914.
T(n+1,n) gives A000217.
T(n+2,n) gives A000914.
T(2n,n) gives A345342.
Cf. A060281, A132393, A186366, A345341.

b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*
      b(n-j)*binomial(n-1, j-1)*ceil(2^(j-2))), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..12);

b[n_] := b[n] = If[n == 0, 1, Sum[Expand[x*b[n-j]*
     Binomial[n-1, j-1]*Ceiling[2^(j-2)]], {j, n}]];
T[n_] := CoefficientList[b[n], x];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Aug 23 2021, after Alois P. Heinz *)

Sum_{k=1..n} k * T(n,k) = A345341(n).

For fixed k, T(n,k) ~ (2*k)^n / (4^k * k!). - Vaclav Kotesovec, Jul 15 2021

cp's OEIS Frontend

A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3*x)^(-1/3), (-1/3)*log(1 - 3*x)). A generalized Stirling1 triangle.

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A052801 A simple grammar: labeled pairs of sequences of cycles.

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A129062 T(n, k) = [x^k] Sum_{k=0..n} Stirling2(n, k)*RisingFactorial(x, k), triangle read by rows, for n >= 0 and 0 <= k <= n.

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A144739 7-factorial numbers A114799(7*n+3): Partial products of A017017(k) = 7*k+3, a(0) = 1.

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A164652 Triangle read by rows: Hultman numbers: a(n,k) is the number of permutations of n elements whose cycle graph (as defined by Bafna and Pevzner) contains k cycles for n >= 0 and 1 <= k <= n+1.

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A285849 Number T(n,k) of permutations of [n] with k ordered cycles such that equal-sized cycles are ordered with increasing least elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A286718 Triangle read by rows: T(n, k) is the Sheffer triangle ((1 - 3x)^(-1/3), (-1/3)log(1 - 3*x)). A generalized Stirling1 triangle.

A144739 7-factorial numbers A114799(7n+3): Partial products of A017017(k) = 7k+3, a(0) = 1.