A098558 -id:A098558 - cp's OEIS frontend

A008290 Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).

1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 9, 8, 6, 0, 1, 44, 45, 20, 10, 0, 1, 265, 264, 135, 40, 15, 0, 1, 1854, 1855, 924, 315, 70, 21, 0, 1, 14833, 14832, 7420, 2464, 630, 112, 28, 0, 1, 133496, 133497, 66744, 22260, 5544, 1134, 168, 36, 0, 1, 1334961, 1334960, 667485, 222480, 55650, 11088, 1890, 240, 45, 0, 1

Offset: 0

N. J. A. Sloane

This is a binomial convolution triangle (Sheffer triangle) of the Appell type: (exp(-x)/(1-x),x), i.e., the e.g.f. of column k is (exp(-x)/(1-x))*(x^k/k!). See the e.g.f. given by V. Jovovic below. - Wolfdieter Lang, Jan 21 2008
The formula T(n,k) = binomial(n,k)*A000166(n-k), with the derangements numbers (subfactorials) A000166 (see also the Charalambides reference) shows the Appell type of this triangle. - Wolfdieter Lang, Jan 21 2008
T(n,k) is the number of permutations of {1,2,...,n} having k pairs of consecutive right-to-left minima (0 is considered a right-to-left minimum for each permutation). Example: T(4,2)=6 because we have 1243, 1423, 4123, 1324, 3124 and 2134; for example, 1324 has right-to-left minima in positions 0-1,3-4 and 2134 has right-to-left minima in positions 0,2-3-4, the consecutive ones being joined by "-". - Emeric Deutsch, Mar 29 2008
T is an example of the group of matrices outlined in the table in A132382--the associated matrix for the sequence aC(0,1). - Tom Copeland, Sep 10 2008
A refinement of this triangle is given by A036039. - Tom Copeland, Nov 06 2012
This triangle equals (A211229(2*n,2*k)) n,k >= 0. - Peter Bala, Dec 17 2014

			exp((y-1)*x)/(1-x) = 1 + y*x + (1/2!)*(1+y^2)*x^2 + (1/3!)*(2 + 3*y + y^3)*x^3 + (1/4!)*(9 + 8*y + 6*y^2 + y^4)*x^4 + (1/5!)*(44 + 45*y + 20*y^2 + 10*y^3 + y^5)*x^5 + ...
Triangle begins:
       1
       0      1
       1      0     1
       2      3     0     1
       9      8     6     0    1
      44     45    20    10    0    1
     265    264   135    40   15    0   1
    1854   1855   924   315   70   21   0  1
   14833  14832  7420  2464  630  112  28  0 1
  133496 133497 66744 22260 5544 1134 168 36 0 1
...
From _Peter Bala_, Feb 13 2017: (Start)
The infinitesimal generator has integer entries given by binomial(n,k)*(n-k-1)! for n >= 2 and 0 <= k <= n-2 and begins
   0
   0  0
   1  0  0
   2  3  0  0
   6  8  6  0 0
  24 30 20 10 0 0
...
It is essentially A238363 (unsigned and omitting the main diagonal), A211603 (with different offset) and appears to be A092271, again without the main diagonal. (End)

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 173, Table 5.2 (without row n=0 and column k=0).
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.
Arnold Kaufmann, Introduction à la combinatorique en vue des applications, Dunod, Paris, 1968. See p. 92.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

Alois P. Heinz, Rows n = 0..150, flattened (first 51 rows from T. D. Noe)
Taha Akbari, Prove using combinatorics Sum_{k=0..n} (k-1)^2 D_n(k)=n!, Mathematics Stack Exchange, Jun 06 2017
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Stefano Capparelli, Margherita Maria Ferrari, Emanuele Munarini, and Norma Zagaglia Salvi, A Generalization of the "Problème des Rencontres", J. Int. Seq. 21 (2018), #18.2.8.
Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016. See Table 1.
S. K. Das and N. Deo, Rencontres graphs: a family of bipartite graphs, Fib. Quart., Vol. 25, No. 3, August 1987, 250-262.
Robert W. Donley Jr, Binomial arrays and generalized Vandermonde identities, arXiv:1905.01525 [math.CO], 2019.
FindStat - Combinatorial Statistic Finder, The number of adjacencies of a permutation, 0 appended, The number of fixed points of a permutation
I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.
J. Liese and J. Remmel, Q-analogues of the number of permutations with k-excedances, PU. M. A. Vol. 21 (2010), No. 2, pp. 285-320 (see E_{n,0}(x) in Table 1 p. 291).
L. Takacs, On the "problème des ménages", Discr. Math. 36 (3) (1981) 289-297, Table 2.
Wikipedia, Rencontres numbers.
Index entries for sequences related to permutations with fixed points

Mirror of triangle A098825.
Columns give A000166, A000240, A000387, A000449, A000475, A129135, A129136, A129149, A129153, A129217, A129218, A129238, A129255.
Cf. A055137, A008291, A098558.
Cf. A080955.
Cf. A000012, A000142 (row sums), A000354.
Cf. A170942. Sub-triangle of A211229.
Cf. A092271, A111492, A211603, A238363.
T(2n,n) gives A281262.
Cf. A000023, A133314, A238385, A320582, A335111.

a008290 n k = a008290_tabl !! n !! k
a008290_row n = a008290_tabl !! n
a008290_tabl = map reverse a098825_tabl
-- Reinhard Zumkeller, Dec 16 2013

T:= proc(n,k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
      (T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Mar 15 2013

a[0] = 1; a[1] = 0; a[n_] := Round[n!/E] /; n >= 1 size = 8; Table[Binomial[n, k]a[n - k], {n, 0, size}, {k, 0, n}] // TableForm (* Harlan J. Brothers, Mar 19 2007 *)
T[n_, k_] := Subfactorial[n-k]*Binomial[n, k]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 12 2017 *)
T[n_, k_] := If[n<1, Boole[n==0 && k==0], T[n, k] = T[n-1, k-1] + T[n-1, k]*(n-1-k) + T[n-1, k+1]*(k+1)]; (* Michael Somos, Sep 13 2024 *)
T[0, 0]:=1; T[n_, 0]:=T[n, 0]=n  T[n-1, 0]+(-1)^n; T[n_, k_]:=T[n, k]=n/k T[n-1, k-1];
Flatten@Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Oliver Seipel, Nov 26 2024 *)

{T(n, k) = if(k<0 || k>n, 0, n!/k! * sum(i=0, n-k, (-1)^i/i!))}; /* Michael Somos, Apr 26 2000 */

T(n, k) = T(n-1, k)*n + binomial(n, k)*(-1)^(n-k) = T(n, k-1)/k + binomial(n, k)*(-1)^(n-k)/(n-k+1) = T(n-1, k-1)*n/k = T(n-k, 0)*binomial(n, k) = A000166(n-k)*binomial(n,k) [with T(0, 0) = 1]; so T(n, n) = 1, T(n, n-1) = 0, T(n, n-2) = n*(n-1)/2 for n >= 0.
Sum_{k=0..n} T(n, k) = Sum_{k=0..n} k * T(n, k) = n! for all n > 0, n, k integers. - Wouter Meeussen, May 29 2001
From Vladeta Jovovic, Aug 12 2002: (Start)
O.g.f. for k-th column: (1/k!)*Sum_{i>=k} i!*x^i/(1+x)^(i+1).
O.g.f. for k-th row: k!*Sum_{i=0..k} (-1)^i/i!*(1-x)^i. (End)
E.g.f.: exp((y-1)*x)/(1-x). - Vladeta Jovovic, Aug 18 2002
E.g.f. for number of permutations with exactly k fixed points is x^k/(k!*exp(x)*(1-x)). - Vladeta Jovovic, Aug 25 2002
Sum_{k=0..n} T(n, k)*x^k is the permanent of the n X n matrix with x's on the diagonal and 1's elsewhere; for x = 0, 1, 2, 3, 4, 5, 6 see A000166, A000142, A000522, A010842, A053486, A053487, A080954. - Philippe Deléham, Dec 12 2003; for x = 1+i see A009551 and A009102. - John M. Campbell, Oct 11 2011
T(n, k) = Sum_{j=0..n} A008290(n, j)*k^(n-j) is the permanent of the n X n matrix with 1's on the diagonal and k's elsewhere; for k = 0, 1, 2 see A000012, A000142, A000354. - Philippe Deléham, Dec 13 2003
T(n,k) = Sum_{j=0..n} (-1)^(j-k)*binomial(j,k)*n!/j!. - Paul Barry, May 25 2006
T(n,k) = (n!/k!)*Sum_{j=0..n-k} ((-1)^j)/j!, 0 <= k <= n. From the Appell type of the triangle and the subfactorial formula.
T(n,0) = n*Sum_{j=0..n-1} (j/(j+1))*T(n-1,j), T(0,0)=1. From the z-sequence of this Sheffer triangle z(j)=j/(j+1) with e.g.f. (1-exp(x)*(1-x))/x. See the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jan 21 2008
T(n,k) = (n/k)*T(n-1,k-1) for k >= 1. See above. From the a-sequence of this Sheffer triangle a(0)=1, a(n)=0, n >= 1 with e.g.f. 1. See the W. Lang link under A006232 for Sheffer a- and z-sequences. - Wolfdieter Lang, Jan 21 2008
From Henk P. J. van Wijk, Oct 29 2012: (Start)
T(n,k) =              T(n-1,k)*(n-1-k) + T(n-1,k+1)*(k+1) for k=0 and
T(n,k) = T(n-1,k-1) + T(n-1,k)*(n-1-k) + T(n-1,k+1)*(k+1) for k>=1.
(End)
T(n,k) = A098825(n,n-k). - Reinhard Zumkeller, Dec 16 2013
Sum_{k=0..n} k^2 * T(n, k) = 2*n! if n > 1. - Michael Somos, Jun 06 2017
From Tom Copeland, Jul 26 2017: (Start)
The lowering and raising operators of this Appell sequence of polynomials P(n,x) are L = d/dx and R = x + d/dL log[exp(-L)/(1-L)] = x-1 + 1/(1-L) = x + L + L^2 - ... such that L P(n,x) = n P(n-1,x) and R P(n,x) = P(n+1,x).
P(n,x) = (1-L)^(-1) exp(-L) x^n = (1+L+L^2+...)(x-1)^n = n! Sum_{k=0..n} (x-1)^k / k!.
The formalism of A133314 applies to the pair of entries A008290 and A055137.
The polynomials of this pair P_n(x) and Q_n(x) are umbral compositional inverses; i.e., P_n(Q.(x)) = x^n = Q_n(P.(x)), where, e.g., (Q.(x))^n = Q_n(x).
For more on the infinitesimal generator, noted by Bala below, see A238385. (End)
Sum_{k=0..n} k^m * T(n,k) = A000110(m)*n! if n >= m. - Zhujun Zhang, May 24 2019
Sum_{k=0..n} (k+1) * T(n,k) = A098558(n). - Alois P. Heinz, Mar 11 2022
From Alois P. Heinz, May 20 2023: (Start)
Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
Sum_{k=0..n} (-1)^k * k * T(n,k) = A335111(n). (End)
T(n,k) = A145224(n,k)+A145225(n,k), refined by even and odd perms. - R. J. Mathar, Jul 06 2023

Comments and more terms from Michael Somos, Apr 26 2000 and Christian G. Bower, Apr 26 2000

A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

Original entry on oeis.org

0, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000, 2248001455555215360000

Offset: 0

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

For n >= 1 a(n) is the size of the centralizer of a transposition in the symmetric group S_(n+1). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 12 2001
For n > 0, a(n) = n! - A062119(n-1) = number of permutations of length n that have two specified elements adjacent. For example, a(4) = 12 as of the 24 permutations, 12 have say 1 and 2 adjacent: 1234, 2134, 1243, 2143, 3124, 3214, 4123, 4213, 3412, 3421, 4312, 4321. - Jon Perry, Jun 08 2003
With different offset, denominators of certain sums computed by Ramanujan.
From Michael Somos, Mar 04 2004: (Start)
Stirling transform of a(n) = [2, 4, 12, 48, 240, ...] is A000629(n) = [2, 6, 26, 150, 1082, ...].
Stirling transform of a(n-1) = [1, 2, 4, 12, 48, ...] is A007047(n-1) = [1, 3, 11, 51, 299, ...].
Stirling transform of a(n) = [1, 4, 12, 48, 240, ...] is A002050(n) = [1, 5, 25, 149, 1081, ...].
Stirling transform of 2*A006252(n) = [2, 2, 4, 8, 28, ...] is a(n) = [2, 4, 12, 48, 240, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 2*A005649(n) = [4, 16, 88, 616, ...].
Stirling transform of a(n+1) = [4, 12, 48, 240, ...] is 4*A083410(n) = [4, 16, 88, 616, ...]. (End)
Number of {12, 12*, 21, 21*}-avoiding signed permutations in the hyperoctahedral group.
Permanent of the (0, 1)-matrices with (i, j)-th entry equal to 0 if and only if it is in the border but not the corners. The border of a matrix is defined the be the first and the last row, together with the first and the last column. The corners of a matrix are the entries (i = 1, j = 1), (i = 1, j = n), (i = n, j = 1) and (i = n, j = n). - Simone Severini, Oct 17 2004

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 520.

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 490.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 817.
Anna Khmelnitskaya, Gerard van der Laan, and Dolf Talmanm, The Number of Ways to Construct a Connected Graph: A Graph-Based Generalization of the Binomial Coefficients, J. Int. Seq. (2023) Art. 23.4.3. See p. 11.
T. Mansour and J. West, Avoiding 2-letter signed patterns, arXiv:math/0207204 [math.CO], 2002.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
Index entries for sequences related to factorial base representation.

Essentially the same sequence as A098558.
Cf. A062119, A080046, A245334, A000142.
Row 3 of A276955 (from term a(2)=4 onward).

a052849 n = if n == 0 then 0 else 2 * a000142 n
a052849_list = 0 : fs where fs = 2 : zipWith (*) [2..] fs
-- Reinhard Zumkeller, Aug 31 2014

[0] cat [2*Factorial(n-1): n in [2..25]]; // Vincenzo Librandi, Nov 03 2014

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

Join[{0}, 2Range[20]!] (* Harvey P. Dale, Jul 13 2013 *)

PARI
```
a(n)=if(n<1,0,n!*2)
```

a(n) = T(n, 2) for n>1, where T is defined as in A080046.
D-finite with recurrence: {a(0) = 0, a(1) = 2, (-1 - n)*a(n+1) + a(n+2)=0}.
E.g.f.: 2*x/(1-x).
a(n) = A090802(n, n - 1) for n > 0. - Ross La Haye, Sep 26 2005
For n >= 1, a(n) = (n+3)!*Sum_{k=0..n+2} (-1)^k*binomial(2, k)/(n + 3 - k). - Milan Janjic, Dec 14 2008
G.f.: 2/Q(0) - 2, where Q(k) = 1 - x*(k + 1)/(1 - x*(k + 1)/Q(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Apr 01 2013
G.f.: -2 + 2/Q(0), where Q(k) = 1 + k*x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 01 2013
G.f.: W(0) - 2 , where W(k) = 1 + 1/( 1 - x*(k+1)/( x*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 21 2013
a(n) = A245334(n, n-1), n > 0. - Reinhard Zumkeller, Aug 31 2014
From Amiram Eldar, Jan 15 2023: (Start)
Sum_{n>=1} 1/a(n) = (e-1)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (e-1)/(2*e). (End)

More terms from Ross La Haye, Sep 26 2005

A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 12, 22, 12, 1, 1, 27, 92, 92, 27, 1, 1, 58, 359, 604, 359, 58, 1, 1, 121, 1311, 3607, 3607, 1311, 121, 1, 1, 248, 4540, 19912, 31238, 19912, 4540, 248, 1, 1, 503, 15110, 102842, 244424, 244424, 102842, 15110, 503, 1

Offset: 0

N. J. A. Sloane

			Triangle begins:
   1;
   1,   1;
   1,   2,    1;
   1,   5,    5,    1;
   1,  12,   22,   12,    1;
   1,  27,   92,   92,   27,    1;
   1,  58,  359,  604,  359,   58,   1;
   1, 121, 1311, 3607, 3607, 1311, 121, 1;
   ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 254.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 215.

Alois P. Heinz, Rows n = 0..140, flattened
Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 22.

Cf. A000007, A008292, A098558 (row sums), A177042 (T(2*n, n)).

Columns include A000325 (k=1).

Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[Eulerian(n, k-1) + Eulerian(n, k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jun 18 2024

t[n_ /; n >= 0, 0] = 1; t[n_, k_] /; k<0 || k>n = 0; t[n_, k_] := t[n, k] = (n-k) t[n-1, k-1] + (k+1) t[n-1, k];
A[n_, k_] /; k == n+1 = 0; A[n_, k_] := t[n, n-k];
T[n_, k_] := A[n, k] + A[n, k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019, after Franck Maminirina Ramaharo *)

def Eulerian(n,k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in range(k+2))
flatten([[Eulerian(n,k-1) + Eulerian(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 18 2024

E.g.f.: (exp(x) - y*exp(y*x))/(exp(y*x) - y*exp(x)). - Vladeta Jovovic, Apr 06 2001
T(n,k) = A123125(n,k) + A123125(n,k+1), with A123125(n,n+1) = 0. - Franck Maminirina Ramaharo, Oct 21 2018
From G. C. Greubel, Jun 18 2024: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} (-1)^k*T(n, k) = A000007(n). (End)

More terms from Vladeta Jovovic, Apr 06 2001

A208529 Number of permutations of n > 1 having exactly 2 points on the boundary of their bounding square.

Original entry on oeis.org

2, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000

Offset: 2

David Nacin, Feb 27 2012

A bounding square for a permutation of n is the square with sides parallel to the coordinate axis containing (1,1) and (n,n), and the set of points P of a permutation p is the set {(k,p(k)) for 0 < k < n+1}.
Sequences A098558 and A052849 have the same terms except for the first. - Joerg Arndt, Mar 03 2012
a(n) is the number of permutations of n symbols that commute with a transposition: a permutation p of {1,...,n} has exactly two points on the boundary of their bounding square if and only if p commutes with transposition (1, n). - Luis Manuel Rivera Martínez, Feb 27 2014
a(n) is also the determinant of a matrix M each of whose elements M(i, j) is the result of a Reverse and Add operation (RADD) on i in base j: M(i,j) = i + (reverse(i) represented in base j), with 1 <= i < n and 1 < j <= n. - Federico Provvedi, May 10 2024

			a(2) = 2 because {(1,1),(2,2)} and {(1,2),(2,1)} each have two points on the bounding square.

Vincenzo Librandi, Table of n, a(n) for n = 2..200
Emeric Deutsch, Permutations and their bounding squares, Math Magazine, 85(1) (2012), 63.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A000142, A098916, A208528.

[2*Factorial(n)/n: n in [1..40]]; // Vincenzo Librandi, Apr 15 2014

A208529:=n->2*(n-2)!; seq(A208529(n), n=2..25); # Wesley Ivan Hurt, Feb 27 2014

Table[2(n-2)!, {n, 2, 10}]
FoldList[Times, 2, Range@21] (* Arkadiusz Wesolowski, May 08 2012 *)
Table[2 n!/n, {n, 1, 40}] (* Vincenzo Librandi, Apr 15 2014 *)

vector(33,n,2*n!/n) /* Anders Hellström, Jul 07 2015 */

import math
def a(n):
    return 2*math.factorial(n-2)

a(n) = 2*(n-2)!.
G.f.: G(0), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
G.f.: 2 + 2*x/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+1)*(2*k+2)/( 1 - 2*x*(2*k+2) - x^2*(2*k+2)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Sep 23 2013
a(n) = 2*n!/(n*(n-1)). - Vincenzo Librandi, Apr 15 2014
E.g.f.: 2 - (1 - x)*(2 + log(1/(1 - x)^2)). - Ilya Gutkovskiy, Mar 21 2018
Sum_{n>=2} 1/a(n) = e/2. - Amiram Eldar, Feb 02 2023

A128564 Triangle, read by rows, where T(n,k) equals the number of permutations of {1..n+1} with [(nk+k)/2] inversions for n>=k>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 9, 22, 15, 1, 1, 29, 90, 90, 29, 1, 1, 49, 359, 573, 359, 98, 1, 1, 174, 1415, 3450, 3450, 1415, 174, 1, 1, 285, 5545, 17957, 29228, 21450, 5545, 628, 1, 1, 1068, 21670, 110010, 230131, 230131, 110010, 21670, 1068, 1

Offset: 0

Paul D. Hanna, Mar 12 2007

Row sums equal 2*n! for n>0.

			Row sums equal 2*n! for n>0:
[1, 2, 4, 12, 48, 240, 1440, 10080, 80640, ..., 2*n!,...].
Triangle begins:
  1;
  1,    1;
  1,    2,     1;
  1,    5,     5,      1;
  1,    9,    22,     15,       1;
  1,   29,    90,     90,      29,       1;
  1,   49,   359,    573,     359,      98,       1;
  1,  174,  1415,   3450,    3450,    1415,     174,      1;
  1,  285,  5545,  17957,   29228,   21450,    5545,    628,     1;
  1, 1068, 21670, 110010,  230131,  230131,  110010,  21670,  1068,    1;
  1, 1717, 84591, 526724, 1729808, 2409581, 1729808, 686763, 84591, 4015, 1;
  ...

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

Cf. A008302 (Mahonian numbers); A128565 (column 1), A128566 (column 2).

Row sums give A098558.

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+
       add(b(u-j, o+j-1)*x^(u-j), j=1..u)))
    end:
T:= (n, k)-> coeff(b(n+1, 0), x, iquo((n+1)*k, 2)):
seq(seq(T(n,k), k=0..n), n=0..10);  # Alois P. Heinz, May 02 2017

b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1, Sum[b[u + j - 1, o - j]* x^(u+j-1), {j, 1, o}] + Sum[b[u-j, o+j-1]*x^(u-j), {j, 1, u}]]];
T[n_, k_] := Coefficient[b[n+1, 0], x, Quotient[(n+1)*k, 2]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{T(n,k)=local(faq=prod(j=1, n+1, (1-q^j)/(1-q))); polcoeff(faq, (n*k+k)\2, q)}

T(n,k) = A008302(n+1, [(nk+k)/2]) = coefficient of q^[(nk+k)/2] in the q-factorial of n+1 for n>=0.

A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.

Original entry on oeis.org

1, 2, 5, 14, 45, 174, 825, 4738, 32137, 251338, 2224157, 21952358, 238962581, 2843085270, 36696680241, 510647009850, 7619901954001, 121367981060434, 2055085325869813, 36861997532438654, 698193329457246653, 13924819967953406654, 291683979376372766697, 6402385486361598687666, 146948520147021794869977

Offset: 1

R. L. Graham, May 02 2017

nonn

Indices of records in A285200.

When prefixed by a(0)=0, the first differences give A111063. - N. J. A. Sloane, May 03 2017

N. J. A. Sloane, Table of n, a(n) for n = 1..400
Ken Knowlton, Illustration showing what floor the elevator is on for the first 49 stages

Cf. A285200, A285203, A111063, A098558, A052849, A286281, A286282.

a:= proc(n) option remember; `if`(n<3, n, ((n-1)^2*a(n-1)
      -(n-2)*(2*n-3)*a(n-2)+(n-1)*(n-3)*a(n-3))/(n-2))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 11 2018

a[n_] := 2 - n + 2 Sum[k!/j!, {k, 0, n-2}, {j, 0, k}];
Array[a, 25] (* Jean-François Alcover, Nov 01 2020 *)

a(n) = 2 - n + 2 * Sum_{k=0..n-2} Sum_{j=0..k} k!/j!.
For n >= 2, a(n) = 1+n+2*Sum_{k=2..n} C(n,k)*(k-1)! = 1+n+2*n!*Sum_{k=2..n} 1/(k*(n-k)!). - N. J. A. Sloane, May 03 2017
E.g.f.: exp(x)*(1-x-2*log(1-x)). Omitting the factor exp(x), this gives (essentially) the e.g.f. for A098558 (or A052849). - N. J. A. Sloane, May 03 2017

A137513 Triangle read by rows: the coefficients of the Mittag-Leffler polynomials.

Original entry on oeis.org

1, 0, 2, 0, 0, 4, 0, 4, 0, 8, 0, 0, 32, 0, 16, 0, 48, 0, 160, 0, 32, 0, 0, 736, 0, 640, 0, 64, 0, 1440, 0, 6272, 0, 2240, 0, 128, 0, 0, 33792, 0, 39424, 0, 7168, 0, 256, 0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512, 0, 0, 2594304, 0, 3676160, 0, 924672, 0, 61440, 0, 1024

Offset: 1

Roger L. Bagula, Apr 23 2008

Previous name was: Triangle read by rows: coefficients of the expansion of a polynomial related to the Poisson kernel: p(t,r) = ((1 + t)/(1 - t))^x: r*Exp(i*theta) -> t.
General relation is that Poisson's kernel is the real part of this type of function (page 31 Hoffman reference).
The row polynomials of this table are the Mittag-Leffler polynomials M(n,t), a polynomial sequence of binomial type [Roman, Chapter 4, Section 1.6]. The first few values are M(0,t) = 1, M(1,t) = 2*t, M(2,t) = 4*t^2, M(3,t) = 4*t+8*t^3. The polynomials M(n,t/2) are the (unsigned) row polynomials of A049218. - Peter Bala, Dec 04 2011
Also the Bell transform of the sequence "a(n) = 2*n! if n is even else 0". For the definition of the Bell transform see A264428. - Peter Luschny, Jan 28 2016

			{1},
{0, 2},
{0, 0, 4},
{0, 4, 0, 8},
{0, 0, 32, 0, 16},
{0, 48, 0, 160, 0, 32},
{0, 0, 736, 0, 640, 0, 64},
{0, 1440, 0, 6272, 0, 2240, 0, 128},
{0, 0, 33792, 0, 39424, 0, 7168, 0, 256},
{0, 80640, 0, 418816, 0, 204288, 0, 21504, 0, 512},
{0, 0, 2594304, 0, 3676160,0, 924672, 0, 61440, 0, 1024}

Kenneth Hoffman, Banach Spaces of Analytic Functions, Dover, New York, 1962, page 30.
Thomas McCullough, Keith Phillips, Foundations of Analysis in the Complex Plane, Holt, Reinhart and Winston, New York, 1973, 215.
S. Roman, The Umbral Calculus: Dover Publications, New York (2005).

Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
H. Bateman, The Polynomial of Mittag-Leffler. PNAS, 26 (8), 1940, 491-496.
Eric Weisstein's World of Mathematics, Mittag-Leffler Polynomial

Cf. A049218, A098558 (row sums).

A137513_row := proc(n) `if`(n=0,1,2*x*hypergeom([1-n,1-x],[2],2));
PolynomialTools[CoefficientList](expand(n!*simplify(%,hypergeom)),x) end:
seq(A137513_row(n),n=0..10): ListTools[FlattenOnce]([%]); # Peter Luschny, Jan 28 2016
# Alternatively, using the function BellMatrix defined in A264428:
BellMatrix(n -> `if`(n::odd, 0, 2*n!), 9); # Peter Luschny, Jan 28 2016

p[t_] = ((1 + t)/(1 - t))^x; Table[ ExpandAll[n! * SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]], {n, 0, 10}]; a = Table[ CoefficientList[n!*SeriesCoefficient[ FullSimplify[Series[p[t], {t, 0, 30} ]], n], x], {n, 0, 10}]; Flatten[a]
MLP[n_] := Sum[Binomial[n, k]*2^k*FactorialPower[n - 1, n - k]* FactorialPower[x, k] // FunctionExpand, {k, 0, n}]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)
MLP[0] = 1; MLP[n_] := 2x*n!*Hypergeometric2F1[1-n, 1-x, 2, 2]; Table[ CoefficientList[MLP[n], x], {n, 0, 9}] // Flatten (* or: *)
BellMatrix[If[OddQ[#], 0, 2*#!]&, 9] (* in triangular matrix form, using Peter Luschny's BellMatrix function defined in A264428 *) (* Jean-François Alcover, Jan 29 2016 *)

MLP = lambda n: sum(binomial(n, k)*2^k*falling_factorial(n-1, n-k)* falling_factorial(x, k) for k in (0..n)).expand()
def A137513_row(n): return MLP(n).list()
for n in (0..9): A137513_row(n) # Peter Luschny, Jan 28 2016

From Peter Bala, Dec 04 2011: (Start)
T(n,k) = (-1)^k*(n-1)!*Sum_{i=k..n} (-2)^i*binomial(n,i)/(i-1)!*|Stirling1(i,k)|.
E.g.f.: Sum_{n>=0} M(n,t)*x^n/n! = exp(t*log((1+x)/(1-x))) = ((1+x)/(1-x))^t = exp(2*t*atanh(x)) = 1 + (2*t)*x + (4*t^2)*x^2/2! + (4*t+8*t^3)*x^3/3! + ....
M(n,t) = (n-1)!*Sum_{k = 1..n} k*2^k*binomial(n,k)*binomial(t,k), for n>=1.
Recurrence relation: M(n+1,t) = 2*t*Sum_{k = 0..floor(n/2)} (n!/(n-2*k)!)* M(n-2*k,t), with M(0,t) = 1.
The o.g.f. for the n-th diagonal of the table is a rational function in t, given by the coefficient of x^n/n! in the expansion (with respect to x) of the compositional inverse (x-t*log((1+x)/(1-x)))^(-1) = x/(1-2*t) + 4*t/(1-2*t)^4*x^3/3! + (48*t+64*t^2)/(1-2*t)^7*x^5/5! + ...; for example, the o.g.f. for the fifth subdiagonal is (48*t+64*t^2)/(1-2*t)^7 = 48*t + 736*t^2 + 6272*t^3+ .... See the Bala link.
(End)
The row polynomials satisfy M(n, t+1) - M(n, t-1) = 2*n*M(n, t)/t. - Peter Bala, Nov 16 2016

Edited and new name by Peter Luschny, Jan 28 2016

cp's OEIS Frontend

A008290 Triangle T(n,k) of rencontres numbers (number of permutations of n elements with k fixed points).

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A052849 a(0) = 0; a(n) = 2*n! (n >= 1).

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A008518 Triangle of Eulerian numbers with rows multiplied by 1 + x.

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A208529 Number of permutations of n > 1 having exactly 2 points on the boundary of their bounding square.

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A128564 Triangle, read by rows, where T(n,k) equals the number of permutations of {1..n+1} with [(nk+k)/2] inversions for n>=k>=0.

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A285201 Stage at which Ken Knowlton's elevator (version 1) reaches floor n for the first time.

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A098559 Expansion of e.g.f. (1+3x)/(1-3x).

A098560 Expansion of e.g.f. (1+4x)/(1-4x).