A111492 -id:A111492 - 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

A007840 Number of factorizations of permutations of n letters into ordered cycles.

Original entry on oeis.org

1, 1, 3, 14, 88, 694, 6578, 72792, 920904, 13109088, 207360912, 3608233056, 68495486640, 1408631978064, 31197601660080, 740303842925184, 18738231641600256, 503937595069600896, 14349899305396086912, 431322634732516137216, 13646841876634025159424

Offset: 0

Arnold Knopfmacher

nonn

a(n) is the number of ways to seat n people at an unspecified number of circular tables and then linearly order the nonempty tables. - Geoffrey Critzer, Mar 18 2009
The terms of this sequence for n >= 1 are the row sums of A008275^2, the unsigned version of A039814. - Peter Bala, Jul 22 2014
Signed sequence is the base for an Appell sequence of polynomials with the e.g.f. e^(x*t)/[log(1+t) + 1] = exp(P(.,x),t) that is the umbral compositional inverse for A238385, reverse of A111492, i.e., umbrally evaluated UP(n,P(.,t))= x^n = P(n,UP(.,t)) where UP(n,t) are the polynomials of A238385. Umbrally evaluated means letting (A(.,t))^n = A(n,t) after substituting A for the independent variable of the polynomial. - Tom Copeland, Nov 15 2014
a(n) is the number of unimodal rooted forests on n labeled nodes (i.e., those forests that avoid the patterns 213 and 312). - Kassie Archer, Aug 30 2018
Number of permutations of [n] where fixed points at index j are j-colored and all other points are unicolored. - Alois P. Heinz, Apr 24 2020

Alois P. Heinz, Table of n, a(n) for n = 0..420
K. Anders and K. Archer, Rooted forests that avoid sets of permutations, arXiv:1607.03046 [math.CO], 2016-2017.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 119.
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013, Digital Object Identifier: 10.1109/SSST.2013.6524939.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 122
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
A. Knopfmacher and J. N. Ridley, Reciprocal sums over partitions and compositions, SIAM J. Discrete Math. 6 (1993), no. 3, 388-399.
Chanchal Kumar and Amit Roy, Integer Sequences and Monomial Ideals, arXiv:2003.10098 [math.CO], 2020.

Cf. A052860. Row sums of unsigned version of A039814.

Cf. A238385, A111492.

a:= proc(n) a(n):= n!*`if`(n=0, 1, add(a(k)/(k!*(n-k)), k=0..n-1)) end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 06 2012

Table[Sum[Abs[StirlingS1[n, k]] k!, {k, 0, n}], {n, 0, 20}] (* Geoffrey Critzer, Mar 18 2009 *)

a(n)=n!*polcoeff(1/(1+log(1-x +x*O(x^n))),n) /* Paul D. Hanna, Jul 19 2006 */

{a(n)=local(CF=1+x*O(x)); for(k=0, n-1, CF=1/((n-k)-((n-k+1)\2)^2*x*CF)); n!*polcoeff(1/(1-x*CF), n)} /* Paul D. Hanna, Jul 19 2006 */

def A007840_list(len):
    f, R, C = 1, [1], [1]+[0]*len
    for n in (1..len):
        f *= n
        for k in range(n, 0, -1):
            C[k] = -C[k-1]*((k-1)/k if k>1 else 1)
        C[0] = sum((-1)^k*C[k] for k in (1..n))
        R.append(C[0]*f)
    return R
print(A007840_list(20)) # Peter Luschny, Feb 21 2016

a(n) = Sum_{k=1..n} k! * s(n, k), s(n, k) = unsigned Stirling number of first kind; E.g.f. 1/(1+log(1-z)).
For n>0, a(n) is the permanent of the n X n matrix with entries a(i, i) = i and a(i, j) = 1 elsewhere. - Philippe Deléham, Dec 09 2003
a(n) = A052860(n)/n for n >= 1.
a(n) = n!*Sum_{k=0..n-1} a(k)/k!/(n-k) for n >= 1 with a(0)=1. - Paul D. Hanna, Jul 19 2006
E.g.f.: B(A(x)) where B(x) = 1/(1-x) and A(x) = log(1/(1-x)). - Geoffrey Critzer, Mar 18 2009
a(n) = D^n(1/(1-x)) evaluated at x = 0, where D is the operator exp(x)*d/dx. Cf. A006252. - Peter Bala, Nov 25 2011
E.g.f.: 1/(1+log(1-x)) = 1/(1 - x/(1 - x/(2 - x/(3 - 4*x/(4 - 4*x/(5 - 9*x/(6 - 9*x/(7 - 16*x/(8 - 16*x/(9 - ...)))))))))), a continued fraction. - Paul D. Hanna, Dec 31 2011
a(n) ~ n! * exp(n)/(exp(1)-1)^(n+1). - Vaclav Kotesovec, Jun 21 2013

Extended June 1995

A135278 Triangle read by rows, giving the numbers T(n,m) = binomial(n+1, m+1); or, Pascal's triangle A007318 with its left-hand edge removed.

Original entry on oeis.org

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 10, 5, 1, 6, 15, 20, 15, 6, 1, 7, 21, 35, 35, 21, 7, 1, 8, 28, 56, 70, 56, 28, 8, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 12, 66, 220, 495, 792, 924, 792

Offset: 0

Zerinvary Lajos, Dec 02 2007

T(n,m) is the number of m-faces of a regular n-simplex.
An n-simplex is the n-dimensional analog of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher, i.e., a set of points such that no m-plane contains more than (m + 1) of them. Such points are said to be in general position.
Reversing the rows gives A074909, which as a linear sequence is essentially the same as this.
From Tom Copeland, Dec 07 2007: (Start)
T(n,k) * (k+1)! = A068424. The comment on permuted words in A068424 shows that T is related to combinations of letters defined by connectivity of regular polytope simplexes.
If T is the diagonally-shifted Pascal matrix, binomial(n+m, k+m), for m=1, then T is a fundamental type of matrix that is discussed in A133314 and the following hold.
The infinitesimal matrix generator is given by A132681, so T = LM(1) of A132681 with inverse LM(-1).
With a(k) = (-x)^k / k!, T * a = [ Laguerre(n,x,1) ], a vector array with index n for the Laguerre polynomials of order 1. Other formulas for the action of T are given in A132681.
T(n,k) = (1/n!) (D_x)^n (D_t)^k Gf(x,t) evaluated at x=t=0 with Gf(x,t) = exp[ t * x/(1-x) ] / (1-x)^2.
[O.g.f. for T ] = 1 / { [ 1 - t * x/(1-x) ] * (1-x)^2 }. [ O.g.f. for row sums ] = 1 / { (1-x) * (1-2x) }, giving A000225 (without a leading zero) for the row sums. Alternating sign row sums are all 1. [Sign correction noted by Vincent J. Matsko, Jul 19 2015]
O.g.f. for row polynomials = [ (1+q)**(n+1) - 1 ] / [ (1+q) -1 ] = A(1,n+1,q) on page 15 of reference on Grassmann cells in A008292. (End)
Given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. The e.g.f. for the row polynomials of A is {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. - Tom Copeland, Aug 21 2008
A007318*A097806 as infinite lower triangular matrices. - Philippe Deléham, Feb 08 2009
Riordan array (1/(1-x)^2, x/(1-x)). - Philippe Deléham, Feb 22 2012
The elements of the matrix inverse are T^(-1)(n,k)=(-1)^(n+k)*T(n,k). - R. J. Mathar, Mar 12 2013
Relation to K-theory: T acting on the column vector (-0,d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168 and also A104712, A111492, and A238363. - Tom Copeland, Apr 11 2014
Number of walks of length p>0 between any two distinct vertices of the complete graph K_(n+2) is W(n+2,p)=(-1)^(p-1)*Sum_{k=0..p-1} T(p-1,k)*(-n-2)^k = ((n+1)^p - (-1)^p)/(n+2) = (-1)^(p-1)*Sum_{k=0..p-1} (-n-1)^k. This is equal to (-1)^(p-1)*Phi(p,-n-1), where Phi is the cyclotomic polynomial when p is an odd prime. For K_3, see A001045; for K_4, A015518; for K_5, A015521; for K_6, A015531; for K_7, A015540. - Tom Copeland, Apr 14 2014
Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-1)^0 + A_1*(x-1)^1 + A_2*(x-1)^2 + ... + A_n*(x-1)^n. This sequence gives A_0, ..., A_n as the entries in the n-th row of this triangle, starting at n = 0. - Derek Orr, Oct 14 2014
See A074909 for associations among this array, the Bernoulli polynomials and their umbral compositional inverses, and the face polynomials of permutahedra and their duals (cf. A019538). - Tom Copeland, Nov 14 2014
From Wolfdieter Lang, Dec 10 2015: (Start)
A(r, n) = T(n+r-2, r-1) = risefac(n,r)/r! = binomial(n+r-1, r), for n >= 1 and r >= 1, gives the array with the number of independent components of a symmetric tensors of rank r (number of indices) and dimension n (indices run from 1 to n). Here risefac(n, k) is the rising factorial.
As(r, n) = T(n+1, r+1) = fallfac(n, r)/r! = binomial(n, r), r >= 1 and n >= 1 (with the triangle entries T(n, k) = 0 for n < k) gives the array with the number of independent components of an antisymmetric tensor of rank r and dimension n. Here fallfac is the falling factorial. (End)
The h-vectors associated to these f-vectors are given by A000012 regarded as a lower triangular matrix. Read as bivariate polynomials, the h-polynomials are the complete homogeneous symmetric polynomials in two variables, found in the compositional inverse of an e.g.f. for A008292, the h-vectors of the permutahedra. - Tom Copeland, Jan 10 2017
For a correlation between the states of a quantum system and the combinatorics of the n-simplex, see Boya and Dixit. - Tom Copeland, Jul 24 2017

			The triangle T(n, k) begins:
   n\k  0  1   2   3   4   5   6   7   8  9 10 11 ...
   0:   1
   1:   2  1
   2:   3  3   1
   3:   4  6   4   1
   4:   5 10  10   5   1
   5:   6 15  20  15   6   1
   6:   7 21  35  35  21   7   1
   7:   8 28  56  70  56  28   8   1
   8:   9 36  84 126 126  84  36   9   1
   9:  10 45 120 210 252 210 120  45  10  1
  10:  11 55 165 330 462 462 330 165  55 11  1
  11:  12 66 220 495 792 924 792 495 220 66 12  1
  ... reformatted by _Wolfdieter Lang_, Mar 23 2015
Production matrix begins
   2   1
  -1   1   1
   1   0   1   1
  -1   0   0   1   1
   1   0   0   0   1   1
  -1   0   0   0   0   1   1
   1   0   0   0   0   0   1   1
  -1   0   0   0   0   0   0   1   1
   1   0   0   0   0   0   0   0   1   1
- _Philippe Deléham_, Jan 29 2014
From _Wolfdieter Lang_, Nov 08 2018: (Start)
Recurrence [_Philippe Deléham_]: T(7, 3) = 2*35 + 35 - 15 - 20 = 70.
Recurrence from Riordan A- and Z-sequences: [1,1,repeat(0)] and [2, repeat(-1, +1)]: From Z: T(5, 0) = 2*5 - 10 + 10 - 5 + 1 = 6. From A: T(7, 3) = 35 + 35 = 70.
Boas-Buck column k=3 recurrence: T(7, 3) = (5/4)*(1 + 5 + 15 + 35) = 70. (End)

G. C. Greubel, Table of n, a(n) for the first 101 rows, flattened
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
L. Boya and K. Dixit, Geometry of density states, arXiv:808.1930 [quant-phy], 2017.
V. Buchstaber, Lectures on Toric Topology, Trends in Mathematics - New Series, Information Center for Mathematical Sciences, Vol. 10, No. 1, 2008. p. 7.
Tom Copeland, Cyclotomic polynomials in combinatorics
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator Part VI on simplices
D. Dugger, A Geometric Introduction to K-Theory [From _Tom Copeland_, Apr 11 2014]
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 12.
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, and Jay Pantone, Restricted Permutations Enumerated by Inversions, arXiv:2406.16403 [cs.DM], 2024. See p. 4.
B. Grünbaum and G. C. Shephard, Convex polytopes, Bull. London Math. Soc. (1969) 1 (3): 257-300.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4 (Added by _Tom Copeland_, Oct 01 2015).
Justin Hughes, Representations Arising from an Action on D-neighborhoods of Cayley Graphs, 2013; slides from a talk.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
Wikipedia, Simplex

Cf. A007318, A014410, A228196.
Cf. Column sequences: A000027, A000217, A000292, A000332, A000389, A000579 - A000582, A001287, A001288, A010965 - A011001, A017713 - A017764.
Cf. A000012, A008292.

for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i) od;

Flatten[Table[CoefficientList[D[1/x ((x + 1) Exp[(x + 1) z] - Exp[z]), {z, k}] /. z -> 0, x], {k, 0, 11}]]
CoefficientList[CoefficientList[Series[1/((1 - x)*(1 - x - x*y)), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Nov 22 2017 *)

for(n=0, 20, for(k=0, n, print1(1/k!*sum(i=0, n, (prod(j=0, k-1, i-j))), ", "))) \\ Derek Orr, Oct 14 2014

Trow = lambda n: sum((x+1)^j for j in (0..n)).list()
for n in (0..10): print(Trow(n)) # Peter Luschny, Jul 09 2019

T(n, k) = Sum_{j=k..n} binomial(j,k) = binomial(n+1, k+1), n >= k >= 0, else 0. (Partial sum of column k of A007318 (Pascal), or summation on the upper binomial index (Graham et al. (GKP), eq. (5.10). For the GKP reference see A007318.) - Wolfdieter Lang, Aug 22 2012
E.g.f.: 1/x*((1 + x)*exp(t*(1 + x)) - exp(t)) = 1 + (2 + x)*t + (3 + 3*x + x^2)*t^2/2! + .... The infinitesimal generator for this triangle has the sequence [2,3,4,...] on the main subdiagonal and 0's elsewhere. - Peter Bala, Jul 16 2013
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 27 2013
T(n,k) = A193862(n,k)/2^k. - Philippe Deléham, Jan 29 2014
G.f.: 1/((1-x)*(1-x-x*y)). - Philippe Deléham, Mar 13 2014
From Tom Copeland, Mar 26 2014: (Start)
[From Copeland's 2007 and 2008 comments]
A) O.g.f.: 1 / { [ 1 - t * x/(1-x) ] * (1-x)^2 } (same as Deleham's).
B) The infinitesimal generator for T is given in A132681 with m=1 (same as Bala's), which makes connections to the ubiquitous associated Laguerre polynomials of integer orders, for this case the Laguerre polynomials of order one L(n,-t,1).
C) O.g.f. of row e.g.f.s: Sum_{n>=0} L(n,-t,1) x^n = exp[t*x/(1-x)]/(1-x)^2 = 1 + (2+t)x + (3+3*t+t^2/2!)x^2 + (4+6*t+4*t^2/2!+t^3/3!)x^3+ ... .
D) E.g.f. of row o.g.f.s: ((1+t)*exp((1+t)*x)-exp(x))/t (same as Bala's).
E) E.g.f. for T(n,k)*a(n-k): {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. For example, for a(k)=2^k, the e.g.f. for the row o.g.f.s is {(2+t) exp[(2+t)x] - 2 exp(2x)}/t.
(End)
From Tom Copeland, Apr 28 2014: (Start)
With different indexing
A) O.g.f. by row: [(1+t)^n-1]/t.
B) O.g.f. of row o.g.f.s: {1/[1-(1+t)*x] - 1/(1-x)}/t.
C) E.g.f. of row o.g.f.s: {exp[(1+t)*x]-exp(x)}/t.
These generating functions are related to row e.g.f.s of A111492. (End)
From Tom Copeland, Sep 17 2014: (Start)
A) U(x,s,t)= x^2/[(1-t*x)(1-(s+t)x)] = Sum_{n >= 0} F(n,s,t)x^(n+2) is a generating function for bivariate row polynomials of T, e.g., F(2,s,t)= s^2 + 3s*t + 3t^2 (Buchstaber, 2008).
B) dU/dt=x^2 dU/dx with U(x,s,0)= x^2/(1-s*x) (Buchstaber, 2008).
C) U(x,s,t) = exp(t*x^2*d/dx)U(x,s,0) = U(x/(1-t*x),s,0).
D) U(x,s,t) = Sum[n >= 0, (t*x)^n L(n,-:xD:,-1)] U(x,s,0), where (:xD:)^k=x^k*(d/dx)^k and L(n,x,-1) are the Laguerre polynomials of order -1, related to normalized Lah numbers. (End)
E.g.f. satisfies the differential equation d/dt(e.g.f.(x,t)) = (x+1)*e.g.f.(x,t) + exp(t). - Vincent J. Matsko, Jul 18 2015
The e.g.f. of the Norlund generalized Bernoulli (Appell) polynomials of order m, NB(n,x;m), is given by exponentiation of the e.g.f. of the Bernoulli numbers, i.e., multiple binomial self-convolutions of the Bernoulli numbers, through the e.g.f. exp[NB(.,x;m)t] = (t/(e^t - 1))^(m+1) * e^(xt). Norlund gave the relation to the factorials (x-1)!/(x-1-n)! = (x-1) ... (x-n) = NB(n,x;n), so T(n,m) = NB(m+1,n+2;m+1)/(m+1)!. - Tom Copeland, Oct 01 2015
From Wolfdieter Lang, Nov 08 2018: (Start)
Recurrences from the A- and Z- sequences for the Riordan triangle (see the W. Lang link under A006232 with references), which are A(n) = A019590(n+1), [1, 1, repeat (0)] and Z(n) = (-1)^(n+1)*A054977(n), [2, repeat(-1, 1)]:
  T(0, 0) = 1, T(n, k) = 0 for n < k, and T(n, 0) = Sum_{j=0..n-1} Z(j)*T(n-1, j), for n >= 1, and T(n, k) = T(n-1, k-1) + T(n-1, k), for n >= m >= 1.
Boas-Buck recurrence for columns (see the Aug 10 2017 remark in A036521 also for references):
  T(n, k) = ((2 + k)/(n - k))*Sum_{j=k..n-1} T(j, k), for n >= 1, k = 0, 1, ..., n-1, and input T(n, n) = 1, for n >= 0, (the BB-sequences are alpha(n) = 2 and beta(n) = 1). (End)
T(n, k) = [x^k] Sum_{j=0..n} (x+1)^j. - Peter Luschny, Jul 09 2019

Edited by Tom Copeland and N. J. A. Sloane, Dec 11 2007

A002104 Logarithmic numbers.

Original entry on oeis.org

0, 1, 3, 8, 24, 89, 415, 2372, 16072, 125673, 1112083, 10976184, 119481296, 1421542641, 18348340127, 255323504932, 3809950977008, 60683990530225, 1027542662934915, 18430998766219336, 349096664728623336, 6962409983976703337, 145841989688186383359, 3201192743180799343844

Offset: 0

N. J. A. Sloane

Prime p divides a(p+1). - Alexander Adamchuk, Jul 05 2006
Also number of lists of elements from {1,..,n} with (1st element) = (smallest element), where a list means an ordered subset (cf. A000262), see also Haskell program. - Reinhard Zumkeller, Oct 26 2010
a(n+1) = p_n(-1)  where p_n(x) is the unique degree-n polynomial such that p_n(k) = A133942(k) for k = 0, 1, ..., n. - Michael Somos, Apr 30 2012
a(n) = A006231(n) + n. - Geoffrey Critzer, Oct 04 2012

			From _Reinhard Zumkeller_, Oct 26 2010: (Start)
a(3) = #{[1], [1,2], [1,2,3], [1,3], [1,3,2], [2], [2,3], [3]} = 8;
a(4) = #{[1], [1,2], [1,2,3], [1,2,3,4], [1,2,4], [1,2,4,3], [1,3], [1,3,2], [1,3,2,4], [1,3,4], [1,3,4,2], [1,4], [1,4,2], [1,4,2,3], [1,4,3], [1,4,3,2], [2], [2,3], [2,3,4], [2,4], [2,4,3], [3], [3,4], [4]} = 24. (End)
G.f. = x + 3*x^2 + 8*x^3 + 24*x^4 + 89*x^5 + 415*x^6 + 2372*x^7 + ...

J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..100 (corrected by Michel Marcus, Jan 19 2019)
J. M. Gandhi, On logarithmic numbers, Math. Student, 31 (1963), 73-83. [Annotated scanned copy]
Mengyao Hu, Eloïc Vallée, Tim Seynnaeve, Patrick Emonts, and Jordi Tura, Characterizing Translation-Invariant Bell Inequalities using Tropical Algebra and Graph Polytopes, arXiv:2407.08783 [quant-ph], 2024. See p. 9.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 116
J. C. Tiernan, An efficient search algorithm to find the elementary circuits of a graph, Commun. ACM, 13 (1970), 722-726.
Index entries for sequences related to logarithmic numbers

Cf. A001338, A006231, A007526, A030297, A133942.

import Data.List (subsequences, permutations)
a002104 = length . filter (\xs -> head xs == minimum xs) .
                   tail . choices . enumFromTo 1
   where choices = concat . map permutations . subsequences
-- Reinhard Zumkeller, Feb 21 2012, Oct 25 2010

a := proc(n) option remember; ifelse(n < 2, n, n*a(n-1) - (n-1)*a(n-2) + 1) end:
seq(a(n), n = 0..23); # Peter Luschny, Dec 05 2023

Table[Sum[Sum[m!/k!,{k,0,m}],{m,0,n-1}],{n,1,30}] (* Alexander Adamchuk, Jul 05 2006 *)
a[n_] = n*(HypergeometricPFQ[{1, 1, 1-n}, {2}, -1]); Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 29 2011 *)

x='x+O('x^99); concat([0], Vec(serlaplace(-log(1-x)*exp(x)))) \\ Altug Alkan, Dec 17 2017

{a(n) = sum(k=0, n-1, binomial(n, k) * (n-k-1)!)}; /* Michael Somos, May 08 2019 */

E.g.f.: -log(1 - x) * exp(x).
a(n) = Sum_{k=1..n} Sum_{i=0..n-k} (n-k)!/i!.
a(n) = Sum_{k=1..n} n(n-1)...(n-k+1)/k = A006231(n) + n - Avi Peretz (njk(AT)netvision.net.il), Mar 24 2001
a(n+1) - a(n) = A000522(n).
a(n) = sum{k=0..n-1, binomial(n, k)*(n-k-1)!}, row sums of A111492. - Paul Barry, Aug 26 2004
a(n) = Sum[Sum[m!/k!,{k,0,m}],{m,0,n-1}]. a(n) = Sum[A000522(m),{m,0,n-1}]. - Alexander Adamchuk, Jul 05 2006
For n > 1, the arithmetic mean of the first n terms is a(n-1) + 1. - Franklin T. Adams-Watters, May 20 2010
a(n) = n * 3F1((1,1,1-n); (2); -1). - Jean-François Alcover, Mar 29 2011
Conjecture: a(n) +(-n-1)*a(n-1) +2*(n-1)*a(n-2) +(-n+2)*a(n-3)=0. - R. J. Mathar, Dec 02 2012
From Emanuele Munarini, Dec 16 2017: (Start)
The generating series A(x) = -exp(x)*log(1-x) satisfies the differential equations:
  (1-x)*A'(x) - (1-x)*A(x) = exp(x)
  (1-x)*A''(x) - (3-2*x)*A'(x) + (2-x)*A(x) = 0.
  From the first one, we have the recurrence reported below by R. R. Forberg. From the second one, we have the recurrence conjectured above. (End)
G.f.: conjecture: T(0)*x/(1-2*x)/(1-x), where T(k) = 1 - x^2*(k+1)^2/(x^2*(k+1)^2 - (1 - 2*x*(k+1))*(1 - 2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 18 2013
a(n) ~ exp(1)*(n-1)!. - Vaclav Kotesovec, Mar 10 2014
a(n) = n*a(n-1) - (n-1)*a(n-2) + 1, a(0) = 0, a(1) = 1. - Richard R. Forberg, Dec 15 2014
a(n) = A007526(n) + A006231(n+1) - A030297(n). - Anton Zakharov, Sep 05 2016
0 =  +a(n)*(+a(n+1) -4*a(n+2) +4*a(n+3) -a(n+4)) +a(n+1)*(+2*a(n+2) -5*a(n+3) +2*a(n+4)) +a(n+2)*(+2*a(n+2) -a(n+3) -a(n+4)) +a(n+3)*(+a(n+3)) for all n>=0. - Michael Somos, May 08 2019
From Peter Bala, Sep 12 2022: (Start)
For n, m >= 0, a(n) - a(n + m) == ( a(1) - a(m) ) (mod m). The sequence {mod(a(1) - a(m+1), m): m >= 1} begins [0, 1, 1, 0, 1, 5, 1, 0, 3, 7, 1, 4, 1, 9, 8, 0, 1, 15, 1, 4, ...].
Conjectures:
1) for n, m >= 0, k >= 2, a(n + m*2^k) - a(n) is divisible by 2^k.
2) for n >= 0, a(n + m*p^k) - a(n) + m*p^(k-1) is divisible by p^k for all positive integers m and k, and for all odd primes p. The particular case n = m = k = 1 is stated in the Comments section by Adamchuk. (End)
a(n) = Integral_{t=0..oo} ((t + 1)^n - 1)/(t*e^t) dt. - Velin Yanev, Apr 13 2024
a(n) = Gamma(n)*(e - ((-1)^n)*Gamma(1 - n, -1)) + hypergeom([1, 1], [2, n + 2], 1)/(n + 1) - polygamma(n) - 1/n + i*Pi for n > 0, where polygamma is the digamma function and the bivariate gamma function is the upper incomplete gamma function. - Velin Yanev, Apr 13 2024

More terms from Larry Reeves (larryr(AT)acm.org), Mar 27 2001

A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

Original entry on oeis.org

1, -1, 2, 2, -3, 3, -6, 8, -6, 4, 24, -30, 20, -10, 5, -120, 144, -90, 40, -15, 6, 720, -840, 504, -210, 70, -21, 7, -5040, 5760, -3360, 1344, -420, 112, -28, 8, 40320, -45360, 25920, -10080, 3024, -756, 168, -36, 9, -362880, 403200, -226800, 86400, -25200, 6048, -1260, 240, -45, 10

Offset: 1

Tom Copeland, Feb 25 2014

Let D=d/dx and [A,B]=A·B-B·A. Then each row corresponds to the coefficients of the operators :xD:^k = x^k D^k in the expansion of the commutator [log(D),:xD:^n]=[-log(x),:xD:^n]=sum(k=0 to n-1, a(n,k) :xD:^k). The e.g.f. is derived from [log(D), exp(t:xD:)]=[-log(x), exp(t:xD:)]= log(1+t)exp(t:xD:), using the shift property exp(t:xD:)f(x)=f((1+t)x).
The reversed unsigned array is A111492.
See the mathoverflow link and link therein to an associated mathstackexchange question for other formulas for log(D). In addition, R_x = log(D) = -log(x) + c - sum[n=1 to infnty, (-1)^n 1/n :xD:^n/n!]=
-log(x) + Psi(1+xD) = -log(x) + c + Ein(:xD:), where c is the Euler-Mascheroni constant, Psi(x), the digamma function, and Ein(x), a breed of the exponential integrals (cf. Wikipedia). The :xD:^k ops. commute; therefore, the commutator reduces to the -log(x) term.
Also the n-th row corresponds to the expansion of d[(xD)!/(xD-n)!]/d(xD) = d[:xD:^n]/d(xD) in the operators :xD:^k, or, equivalently, the coefficients of x in  d[z!/(z-n)!]/dz=d[St1(n,z)]]/dz evaluated umbrally with z=St2(.,x), i.e., z^n replaced by St2(n,x), where St1(n,x) and St2(n,x) are the signed and unsigned Stirling polynomials of the first (A008275) and second (A008277) kinds. The derivatives of the unsigned St1 are A028421. See examples. This formalism follows from the relations between the raising and lowering operators presented in the MathOverflow link and the Pincherle derivative. The results can be generalized through the operator relations in A094638, which are related to the celebrated Witt Lie algebra and pseudodifferential operators / symbols, to encompass other integral arrays.
A002741(n)*(-1)^(n+1) (row sums), A002104(n)*(-1)^(n+1) (alternating row sums). Column sequences: A133942(n-1), A001048(n-1), A238474, ... - Wolfdieter Lang, Mar 01 2014
Add an additional head row of zeros to the lower triangular array and denote it as T (with initial indexing in columns and rows being 0). Let dP = A132440, the infinitesimal generator for the Pascal matrix, and I, the identity matrix, then exp(T)=I+dP, i.e., T=log(I+dP). Also, (T_n)^n=0, where T_n denotes the n X n submatrix, i.e., T_n is nilpotent of order n. - Tom Copeland, Mar 01 2014
Any pair of lowering and raising ops. L p(n,x) = n·p(n-1,x) and R p(n,x) = p(n+1,x) satisfy [L,R]=1 which implies (RL)^n = St2(n,:RL:), and since (St2(·,u))!/(St2(·,u)-n)!= u^n, when evaluated umbrally, d[(RL)!/(RL-n)!]/d(RL) = d[:RL:^n]/d(RL) is well-defined and gives A238363 when the LHS is reduced to a sum of :RL:^k terms, exactly as for L=d/dx and R=x above. (Note that R_x above is a raising op. different from x, with associated L_x=-xD.) - Tom Copeland, Mar 02 2014
For relations to colored forests, disposition of flags on flagpoles, and the colorings of the vertices of the complete graphs K_n, encoded in their chromatic polynomials, see A130534. - Tom Copeland, Apr 05 2014
The unsigned triangle, omitting the main diagonal, gives A211603. See also A092271. Related to the infinitesimal generator of A008290. - Peter Bala, Feb 13 2017

			The first few row polynomials are
p(1,x)=  1
p(2,x)= -1 + 2x
p(3,x)=  2 - 3x + 3x^2
p(4,x)= -6 + 8x - 6x^2 + 4x^3
p(5,x)= 24 -30x +20x^2 -10x^3 + 5x^4
...........
For n=3: z!/(z-3)!=z^3-3z^2+2z=St1(3,z) with derivative 3z^2-6z+2, and
3·St2(2,x)-6·St2(1,x)+2=3(x^2+x)-6x+2=3x^2-3x+2=p(3,x). To see the relation to the operator formalism, note that (xD)^k=St2(k,:xD:) and (xD)!/(xD-k)!=[St2(·,:xD:)]!/[St2(·,:xD:)-k]!= :xD:^k.
The triangle a(n,k) begins:
n\k       0       1       2      3      4     5      6    7   8   9 ...
1:        1
2:       -1       2
3:        2      -3       3
4:       -6       8      -6      4
5:       24     -30      20    -10      5
6:     -120     144     -90     40    -15     6
7:      720    -840     504   -210     70   -21      7
8:    -5040    5760   -3360   1344   -420   112    -28    8
9:    40320  -45360   25920 -10080   3024  -756    168  -36   9
10: -362880  403200 -226800  86400 -25200  6048  -1260  240 -45  10
... formatted by _Wolfdieter Lang_, Mar 01 2014
-----------------------------------------------------------------------

Tom Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator, 2014.
Tom Copeland, Bernoulli, Blissard, and Lie meet Stirling and the simplices: State number operators and normal ordering, 2014.
Tom Copeland, Compositional inverse operators and Sheffer sequences, 2016.

Cf. A094587, A132013, A132014, A132159, A235706, A238385.

Cf. A008290, A092271, A211603.

a[n_, k_] := (-1)^(n-k-1)*n!/((n-k)*k!); Table[a[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 09 2015 *)

a(n,k) = (-1)^(n-k-1)*n!/((n-k)*k!) for k=0 to (n-1).
E.g.f.: log(1+t)*exp(x*t).
E.g.f.for unsigned array: -log(1-t)*exp(x*t).
The lowering op. for the row polynomials is L=d/dx, i.e., L p(n,x) = n*p(n-1,x).
An e.g.f. for an unsigned related version is -log(1+t)*exp(x*t)/t= exp(t*s(·,x)) with s(n,x)=(-1)^n * p(n+1,-x)/(n+1). Let L=d/dx and R= x-(1/((1-D)log(1-D))+1/D),then R s(n,x)= s(n+1,x) and L s(n,x)= n*s(n-1,x), defining a special Sheffer sequence of polynomials, an Appell sequence. So, R (-1)^(n-1) p(n,-x)/n = (-1)^n p(n+1,-x)/(n+1).
From Tom Copeland, Apr 17 2014: (Start)
Dividing each diagonal by its first element (-1)^(n-1)*(n-1)! yields the reverse of A104712.
Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result as A007318(x) = P(x). Then with dP = A132440, M = padded A238363 = A238385-I, I = identity matrix, and (B(.,x))^n = B(n,x) = the n-th Bell polynomial Bell(n,x) of A008277,
A) P(x)= exp(x*dP) = exp[x*(e^M-I)] = exp[M*B(.,x)] = (I+dP)^B(.,x), and
B) P(:xD:)=exp(dP:xD:)=exp[(e^M-I):xD:]=exp[M*B(.,:xD:)]=exp[M*xD]=
  (1+dP)^(xD) with action P(:xD:)g(x) = exp(dP:xD:)g(x) = g[(I+dP)*x].
C) P(x)^m = P(m*x). P(2x) = A038207(x) = exp[M*B(.,2x)], face vectors of n-D hypercubes. (End)
From Tom Copeland, Apr 26 2014: (Start)
M = padded A238363 = A238385-I
A)  = [St1]*[dP]*[St2] = [padded A008275]*A132440*A048993
B)  = [St1]*[dP]*[St1]^(-1)
C)  = [St2]^(-1)*[dP]*[St2]
D)  = [St2]^(-1)*[dP]*[St1]^(-1),
  where [St1]=padded A008275 just as [St2]=A048993=padded A008277.
E) P(x) = [St2]*exp(x*M)*[St1] = [St2]*(I + dP)^x*[St1].
F) exp(x*M) = [St1]*P(x)*[St2] = (I + dP)^x,
  where (I + dP)^x = sum(k>=0, C(x,k)*dP^k).
Let the row vector Rv=(c0 c1 c2 c3 ...) and the column vector Cv(x)=(1 x x^2 x^3 ...)^Transpose. Form the power series V(x)= Rv * Cv(x) and W(y) := V(x.) evaluated umbrally with (x.)^n = x_n = (y)_n = y!/(y-n)!. Then
G) U(:xD:) = dV(:xD:)/d(xD) = dW(xD)/d(xD) evaluated with (xD)^n = Bell(n,:xD:),
H) U(x) = dV(x.)/dy := dW(y)/dy evaluated with y^n=y_n=Bell(n,x), and
I) U(x) = Rv * M * Cv(x).   (Cf. A132440, A074909.) (End)
The Bernoulli polynomials Ber_n(x) are related to the polynomials q_n(x) = p(n+1,x) / (n+1) with the e.g.f. [log(1+t)/t] e^(xt) (cf. s_n (x) above) as Ber_n(x) = St2_n[q.(St1.(x))], umbrally, or [St2]*[q]*[St1], in matrix form. Since q_n(x) is an Appell sequence of polynomials, q_n(x) = [log(1+D_x)/D_x]x^n. - Tom Copeland, Nov 06 2016

Pincherle formalism added by Tom Copeland, Feb 27 2014

A104712 Pascal's triangle, with the first two columns removed.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 10, 10, 5, 1, 15, 20, 15, 6, 1, 21, 35, 35, 21, 7, 1, 28, 56, 70, 56, 28, 8, 1, 36, 84, 126, 126, 84, 36, 9, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 78, 286, 715

Offset: 2

Gary W. Adamson, Mar 19 2005

A000295 (Eulerian numbers) gives the row sums.
Write A004736 and Pascal's triangle as infinite lower triangular matrices A and B; then A*B is this triangle.
From Peter Luschny, Apr 10 2011: (Start)
A slight variation has a combinatorial interpretation: remove the last column and the second one from Pascal's triangle. Let P(m, k) denote the set partitions of {1,2,..,n} with the following properties:
(a) Each partition has at least one singleton block;
(c) k is the size of the largest block of the partition;
(b) m = n - k + 1 is the number of parts of the partition.
Then A000295(n) = Sum_{k=1..n} card(P(n-k+1,k)).
For instance, A000295(4) = P(4,1) + P(3,2) + P(2,3) + P(1,4) = card({1|2|3|4}) + card({1|2|34, 1|3|24,1|4|23, 2|3|14, 2|4|13, 3|4|12}) + card({1|234, 2|134, 3|124, 4|123}) = 1 + 6 + 4 = 11.
This interpretation can be superimposed on the sequence by changing the offset to 1 and adding the value 1 in front. The triangle then starts
  1;
  1,  3;
  1,  6,  4;
  1, 10, 10,  5;
  1, 15, 20, 15,  6;
  ...
(End)
Diagonal sums are A001924(n+1). - Philippe Deléham, Jan 11 2014
Relation to K-theory: T acting on the column vector (d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168, A111492, A238363, and A135278. - Tom Copeland, Apr 11 2014

			The triangle a(n, k) begins:
  n\k  2   3   4    5    6    7    8   9  10 11 12 13
  2:   1
  3:   3   1
  4:   6   4   1
  5:  10  10   5    1
  6:  15  20  15    6    1
  7:  21  35  35   21    7    1
  8:  28  56  70   56   28    8    1
  9:  36  84 126  126   84   36    9   1
  10: 45 120 210  252  210  120   45  10   1
  11: 55 165 330  462  462  330  165  55  11  1
  12: 66 220 495  792  924  792  495 220  66 12  1
  13: 78 286 715 1287 1716 1716 1287 715 286 78 13  1
... reformatted. - _Wolfdieter Lang_, Mar 20 2015

G. C. Greubel, Rows n=2..100 of triangle, flattened
D. Dugger, A Geometric Introduction to K-Theory
Candice A. Marshall, Construction of Pseudo-Involutions in the Riordan Group, Dissertation, Morgan State University, 2017.
T. Saito, The discriminant and the determinant of a hypersurface of even dimension (p. 4), arXiv:1110.1717 [math.AG], 2011-2012.

Cf. A000295, A007318, A008292, A104713, A027641/A027642 (first Bernoulli numbers B-), A164555/A027642 (second Bernoulli numbers B+), A176327/A176289.

/* As triangle */ [[Binomial(n, k): k in [2..n]]: n in [2..10]]; // G. C. Greubel, May 15 2018

t[n_, k_] := Binomial[n, k]; Table[ t[n, k], {n, 2, 13}, {k, 2, n}] // Flatten (* Robert G. Wilson v, Apr 16 2011 *)

for(n=2, 10, for(k=2,n, print1(binomial(n,k), ", "))) \\ G. C. Greubel, May 15 2018

T(n,k) = binomial(n,k), for 2 <= k <= n.
From Peter Bala, Jul 16 2013: (Start)
The following remarks assume an offset of 0.
Riordan array (1/(1 - x)^3, x/(1 - x)).
O.g.f.: 1/(1 - t)^2*1/(1 - (1 + x)*t) = 1 + (3 + x)*t + (6 + 4*x + x^2)*t^2 + ....
E.g.f.: (1/x*d/dt)^2 (exp(t)*(exp(x*t) - 1 - x*t)) = 1 + (3 + x)*t + (6 + 4*x + x^2)*t^2/2! + ....
The infinitesimal generator for this triangle has the sequence [3,4,5,...] on the main subdiagonal and 0's elsewhere. (End)
As triangle T(n,k), 0<=k<=n: T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014
From Tom Copeland, Apr 11 2014: (Start)
A) The infinitesimal generator for this matrix is given in A132681 with m=2. See that entry for numerous relations to differential operators and the Laguerre polynomials of order m=2, i.e., Lag(n,t,2) = Sum_{j=0..n} binomial(n+2,n-j)*(-t)^j/j!.
B) O.g.f.: 1 / { [ 1 - t * x/(1-x) ] * (1-x)^3 }
C) O.g.f. of row e.g.f.s: exp[t*x/(1-x)]/(1-x)^3 = [Sum_{n>=0} x^n * Lag(n,-t,2)] = 1 + (3 + t)*x + (6 + 4t + t^2/2!)*x^2 + (10 + 10t + 5t^2/2! + t^3/3!)*x^3 + ....
D) E.g.f. of row o.g.f.s: [(1+t)*exp((1+t)*x) - (1+t+t*x)exp(x)]/t^2. (End)
O.g.f. for m-th row (m=n-2): [(1+x)^(m+2)-(1+(m+2)*x)]/x^2. - Tom Copeland, Apr 16 2014
Reverse T = [St2]*dP*[St1]- dP = [St2]*(exp(x*M)-I)*[St1]-(exp(x*M)-I) with top two rows of zeros removed, [St1]=padded A008275 just as [St2]=A048993=padded A008277, dP= A132440, M=A238385-I, and I=identity matrix. Cf. A238363. - Tom Copeland, Apr 26 2014
O.g.f. of column k (with k leading zeros): (x^k)/(1-x)^(k+1), k >= 2. - Wolfdieter Lang, Mar 20 2015

Edited and extended by David Wasserman, Jul 03 2007

A208535 Square array read by descending antidiagonals: T(n,k) is the number of n-bead necklaces of k colors not allowing reversal, with no adjacent beads having the same color (n, k >= 1).

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 3, 0, 0, 5, 6, 2, 1, 0, 6, 10, 8, 6, 0, 0, 7, 15, 20, 24, 6, 1, 0, 8, 21, 40, 70, 48, 14, 0, 0, 9, 28, 70, 165, 204, 130, 18, 1, 0, 10, 36, 112, 336, 624, 700, 312, 36, 0, 0, 11, 45, 168, 616, 1554, 2635, 2340, 834, 58, 1, 0, 12, 55, 240, 1044, 3360, 7826, 11160

Offset: 1

R. H. Hardin, Feb 27 2012

For prime rows, these appear to be evaluations of Moreau's necklace polynomials at the integers with several combinatorial interpretations (see Wikipedia link). - Tom Copeland, Oct 20 2014
From Petros Hadjicostas, Nov 05 2017: (Start)
The g.f. for column k follows easily from I. Gessel's formulas for this sequence. Since S(1,k) = k-1, we have T(1,k) = k + S(1,k) - (k - 1). Thus, Sum_{n >= 1} T(n,k)*x^n = k*x + Sum_{n >= 1} (1/n)*Sum_{d|n} (k - 1)^d*phi(n/d)*x^n - Sum_{s=0} (k-1)*x^{2*s+1}. Letting m = n/d, we get that (column k g.f.) = k*x - (k - 1)*x/(1 -x^2) + Sum_{m >= 1} (phi(m)/m)*Sum_{d >= 1}((k - 1)*x^m)^d/d. But Sum_{d>=1} z^d/d = -log(1 - z), and so (column k g.f.) = k*x - (k - 1)*x/(1 - x^2) - Sum_{m >= 1} (phi(m)/m)*log(1 - (k - 1)*x^m).
The other formula for the g.f. of column k of this sequence follows from the formula Sum_{n >= 1} (phi(n)/n)*log(1 + t^n) = t/(1 - t^2), which in turn follows from the well-known series Sum_{n >= 1} phi(n)*t^n/(1 + t^n) = t*(1 + t^2)/(1 - t^2)^2.
The extra term k*x in the g.f. for column k is due to the fact that we conventionally assume that a necklace with only one bead, colored with one of the k colors available, is such that there are "no adjacent beads having the same color" (even though, strictly speaking, a single bead is adjacent to itself when we go around the circle of the necklace).
One can use the g.f. for column k to derive the so-called "Empirical for row n" formulae that are denoted by a(k) and given in the formula section below (from n = 1 to n = 7). For example, for n = 3, a(k) = a(k, x=0), where a(k, x) = (1/3!)*d^3/dx^3 (column k g.f.). Here, d^3/dx^3 stands for "third derivative w.r.t. x". If we let f(x) = x/(1 - x^2) and g(x, m) = -log(1 - (k - 1)*x^m), then f^{(3)}(0) = 6, while g^{(3)}(0,m) = 2*(k - 1)^3 for m = 1, 0 for m=2, 6*(k - 1) for m = 3, and 0 for m >= 4. Then, a(k) = (1/6)*(-6*(k - 1) + 2*(k - 1)^3 + (2/3)*6*(k - 1)) = (1/3)*k^3 - k^2 + (2/3)*k. Using this method, one can derive an "Empirical for row n" formula for a(k) for any positive integer n. (End)

			Table T(n,k) (with rows n >= 1 and columns k >= 1) starts:
  1 2  3   4    5     6      7      8       9      10       11       12       13 ...
  0 1  3   6   10    15     21     28      36      45       55       66       78 ...
  0 0  2   8   20    40     70    112     168     240      330      440      572 ...
  0 1  6  24   70   165    336    616    1044    1665     2530     3696     5226 ...
  0 0  6  48  204   624   1554   3360    6552   11808    19998    32208    49764 ...
  0 1 14 130  700  2635   7826  19684   43800   88725   166870   295526   498004 ...
  0 0 18 312 2340 11160  39990 117648  299592  683280  1428570  2783880  5118828 ...
  0 1 36 834 8230 48915 210126 720916 2097684 5381685 12501280 26796726 53750346 ...
  ...
All solutions for n = 4 and k = 3:
  1    2    1    1    1    1
  3    3    2    2    3    2
  2    2    3    1    1    1
  3    3    2    2    3    3

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 313 terms from R. H. Hardin)
MathOverflow, A number array related to colored necklaces and the primes, 2014.
Wikipedia, Necklace polynomial and p-derivation.

Columns 3..6: A106365, A106366, A106367, A106368.
Rows 2..7: A000217(n-1), A007290, A006528(n-1), A208536, A006565(n-1), A208537.
Cf. A111492, A185651, A238363.

T[n_, k_] := If[n == 1, k, Sum[ EulerPhi[n/d]*(k-1)^d, {d, Divisors[n]}]/n - If[OddQ[n], k-1, 0]]; Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)

T(n,k) = if(n==1, k, sumdiv(n,d,eulerphi(n/d)*(k-1)^d)/n - if(n%2, k-1));
for(n=1, 10, for(k=1, 10, print1(T(n,k), ", ")); print) \\ Andrew Howroyd, Oct 14 2017

Let S(n,k) = (1/n) Sum_{d|n} (k-1)^d phi(n/d), where phi is Euler's function.
Then for n even, T(n,k) = S(n,k) and for n > 1 and odd, T(n,k) = S(n,k) - (k-1), and T(1,k) = k. - Ira M. Gessel, Oct 21 2014, Sep 25 2017
Empirical for row n:
n=1: a(k) = k
n=2: a(k) = (1/2)*k^2 - (1/2)*k
n=3: a(k) = (1/3)*k^3 - k^2 + (2/3)*k
n=4: a(k) = (1/4)*k^4 - k^3 + (7/4)*k^2 - k
n=5: a(k) = (1/5)*k^5 - k^4 + 2*k^3 - 2*k^2 + (4/5)*k
n=6: a(k) = (1/6)*k^6 - k^5 + (5/2)*k^4 - (19/6)*k^3 + (7/3)*k^2 - (5/6)*k
n=7: a(k) = (1/7)*k^7 - k^6 + 3*k^5 - 5*k^4 + 5*k^3 - 3*k^2 + (6/7)*k
-----------
From Tom Copeland, Oct 20 2014: (Start)
The first three numbers in each row of the triangular array are given by T(n,k) = (1/k)*(n-k+1)! / (n-2*k+1)!.
For the table here, the first three rows, aside from initial zeros, are given by a(n,k) = (1/n)*(k + 1 - n)! / (k + 1 - 2*n)! or, with no leading zeros, by a(n,k) = (1/n)*(n+k-1)! / (k-1)!. The first three elements of each column correspond to the last three elements of a row in A238363 and the first three of A111492.
Prime rows (> 1) appear to be a(m,n) = (n^m - n)/m. See Wikipedia link. (End)
G.f. for column k: Sum_{n >= 1} T(n,k)*x^n = k*x - Sum_{n >= 1} (phi(n)/n)*((k - 1)*log(1 + x^n) + log(1 - (k - 1)*x^n)) = k*x - (k - 1)*x/(1 - x^2) - Sum_{n >= 1} (phi(n)/n)*log(1 - (k - 1)*x^n). - Petros Hadjicostas, Nov 05 2017

Name edited by Petros Hadjicostas, Jun 24 2020

A092271 Triangle read by rows. First in a series of triangular arrays counting permutations of partitions.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 8, 6, 1, 24, 30, 20, 10, 1, 120, 144, 90, 40, 15, 1, 720, 840, 504, 210, 70, 21, 1, 5040, 5760, 3360, 1344, 420, 112, 28, 1, 40320, 45360, 25920, 10080, 3024, 756, 168, 36, 1, 362880, 403200, 226800, 86400, 25200, 6048, 1260, 240, 45, 1, 3628800, 3991680, 2217600, 831600, 237600, 55440, 11088, 1980, 330, 55, 1

Offset: 1

Alford Arnold, Feb 14 2004

Generate signatures in accordance with A086141. Map to partitions in accordance with A025487. Calculate the number of permutations in accordance with Abramowitz and Stegun, p. 831 (reference M2). Display the results as illustrated by A090774. The second array is:
    3
   20  15
   90 120  45
  504 630 420 105
  ...
Apart from the main diagonal, appears to be the same as A211603 (see also A238363) and is related to the infinitesimal generator of A008290; if so, the off-diagonal triangle entries are given by binomial(n,k)*(n-k-1)! for n >= 2 and 0 <= k <= n-2. - Peter Bala, Feb 13 2017
Let aut(p) denote the size of the centralizer of the partition p (see A339016 for the definition). Then row(n) = [n!/aut(p) for p in P], where P are the partitions of n with largest part k and length n + 1 - k. Row sums are A121726. - Peter Luschny, Nov 19 2020

			The triangle begins:
1:    1
2:    1   1
3:    2   3  1
4:    6   8  6  1
5:   24  30 20 10  1
6:  120 144 90 40 15 1
  ...
From _Peter Luschny_, Nov 19 2020: (Start):
The combinatorial interpretation is illustrated by this computation of row 6:
6! / aut([6])                = 720 / A339033(6, 1) = 720/6   = 120 = T(6, 1)
6! / aut([5, 1])             = 720 / A339033(6, 2) = 720/5   = 144 = T(6, 2)
6! / aut([4, 1, 1])          = 720 / A339033(6, 3) = 720/8   =  90 = T(6, 3)
6! / aut([3, 1, 1, 1])       = 720 / A339033(6, 4) = 720/18  =  40 = T(6, 4)
6! / aut([2, 1, 1, 1, 1])    = 720 / A339033(6, 5) = 720/48  =  15 = T(6, 5)
6! / aut([1, 1, 1, 1, 1, 1]) = 720 / A339033(6, 6) = 720/720 =   1 = T(6, 6)
-------------------------------------------------------------------------------
                                                         Sum:  410 = A121726(6)
(End)

Abramowitz and Stegun, p. 831.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A007290, A025487, A086141, A090774, A008290, A111492, A211603, A238363, A121726, A339016, A339033.

f[list_] :=Total[list]!/Apply[Times, list]/Apply[Times, Map[Length, Split[list]]!];Table[Append[Map[f, Select[Partitions[n], Count[#, Except[1]] == 1 &]], 1], {n,1, 10}] // Grid (* Geoffrey Critzer, Nov 07 2015 *)

def A092271(n, k):
    if n == k: return 1
    return factorial(n) // ((n + 1 - k)*factorial(k - 1))
for n in (1..9): print(n, [A092271(n, k) for k in (1..n)])
def A092271Row(n):
    if n == 0: return [1]
    f = factorial(n); S = []
    for k in range(n,0,-1):
        for p in Partitions(n, max_part=k, inner=[k], length=n+1-k):
            S.append(f // p.aut())
    return S
for n in (1..9): print(A092271Row(n)) # Peter Luschny, Nov 20 2020

More terms from Geoffrey Critzer, Nov 10 2015

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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A007840 Number of factorizations of permutations of n letters into ordered cycles.

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A135278 Triangle read by rows, giving the numbers T(n,m) = binomial(n+1, m+1); or, Pascal's triangle A007318 with its left-hand edge removed.

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A002104 Logarithmic numbers.

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A238363 Coefficients for the commutator for the logarithm of the derivative operator [log(D),x^n D^n]=d[(xD)!/(xD-n)!]/d(xD) expanded in the operators :xD:^k.

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A104712 Pascal's triangle, with the first two columns removed.

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