A021009 -id:A021009 - cp's OEIS frontend

A105278 Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.

1, 2, 1, 6, 6, 1, 24, 36, 12, 1, 120, 240, 120, 20, 1, 720, 1800, 1200, 300, 30, 1, 5040, 15120, 12600, 4200, 630, 42, 1, 40320, 141120, 141120, 58800, 11760, 1176, 56, 1, 362880, 1451520, 1693440, 846720, 211680, 28224, 2016, 72, 1, 3628800, 16329600

Offset: 1

Miklos Kristof, Apr 25 2005

T(n,k) is the number of partially ordered sets (posets) on n elements that consist entirely of k chains. For example, T(4, 3)=12 since there are exactly 12 posets on {a,b,c,d} that consist entirely of 3 chains. Letting ab denote a<=b and using a slash "/" to separate chains, the 12 posets can be given by a/b/cd, a/b/dc, a/c/bd, a/c/db, a/d/bc, a/d/cb, b/c/ad, b/c/da, b/d/ac, b/d/ca, c/d/ab and c/d/ba, where the listing of the chains is arbitrary (e.g., a/b/cd = a/cd/b =...cd/b/a). - Dennis P. Walsh, Feb 22 2007
Also the matrix product |S1|.S2 of Stirling numbers of both kinds.
This Lah triangle is a lower triangular matrix of the Jabotinsky type. See the column e.g.f. and the D. E. Knuth reference given in A008297. - Wolfdieter Lang, Jun 29 2007
The infinitesimal matrix generator of this matrix is given in A132710. See A111596 for an interpretation in terms of circular binary words and generalized factorials. - Tom Copeland, Nov 22 2007
Three combinatorial interpretations: T(n,k) is (1) the number of ways to split [n] = {1,...,n} into a collection of k nonempty lists ("partitions into sets of lists"), (2) the number of ways to split [n] into an ordered collection of n+1-k nonempty sets that are noncrossing ("partitions into lists of noncrossing sets"), (3) the number of Dyck n-paths with n+1-k peaks labeled 1,2,...,n+1-k in some order. - David Callan, Jul 25 2008
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 = D where D(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. - Tom Copeland, Aug 21 2008
An e.g.f. for the row polynomials of A(n,k) = T(n,k)*a(n-k) is exp[a(.)* D_x * x^2] exp(x*t) = exp(x*t) exp[(.)!*Lag(.,-x*t,1)*a(.)*x], umbrally, where [(.)! Lag(.,x,1)]^n = n! Lag(n,x,1) is a normalized Laguerre polynomial of order 1. - Tom Copeland, Aug 29 2008
Triangle of coefficients from the Bell polynomial of the second kind for f = 1/(1-x). B(n,k){x1,x2,x3,...} = B(n,k){1/(1-x)^2,...,(j-1)!/(1-x)^j,...} = T(n,k)/(1-x)^(n+k). - Vladimir Kruchinin, Mar 04 2011
The triangle, with the row and column offset taken as 0, is the generalized Riordan array (exp(x), x) with respect to the sequence n!*(n+1)! as defined by Wang and Wang (the generalized Riordan array (exp(x), x) with respect to the sequence n! is Pascal's triangle A007318, and with respect to the sequence n!^2 is A021009 unsigned). - Peter Bala, Aug 15 2013
For a relation to loop integrals in QCD, see p. 33 of Gopakumar and Gross and Blaizot and Nowak. - Tom Copeland, Jan 18 2016
Also the Bell transform of (n+1)!. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016
Also the number of k-dimensional flats of the n-dimensional Shi arrangement. - Shuhei Tsujie, Apr 26 2019
The numbers T(n,k) appear as coefficients when expanding the rising factorials (x)^k = x(x+1)...(x+k-1) in the basis of falling factorials (x)k = x(x-1)...(x-k+1). Specifically, (x)^n = Sum{k=1..n} T(n,k) (x)k. - _Jeremy L. Martin, Apr 21 2021

			T(1,1) = C(1,1)*0!/0! = 1,
T(2,1) = C(2,1)*1!/0! = 2,
T(2,2) = C(2,2)*1!/1! = 1,
T(3,1) = C(3,1)*2!/0! = 6,
T(3,2) = C(3,2)*2!/1! = 6,
T(3,3) = C(3,3)*2!/2! = 1,
Sheffer a-sequence recurrence: T(6,2)= 1800 = (6/3)*120 + 6*240.
B(n,k) =
   1/(1-x)^2;
   2/(1-x)^3,  1/(1-x)^4;
   6/(1-x)^4,  6/(1-x)^5,  1/(1-x)^6;
  24/(1-x)^5, 36/(1-x)^6, 12/(1-x)^7, 1/(1-x)^8;
The triangle T(n,k) begins:
  n\k      1       2       3      4      5     6    7  8  9 ...
  1:       1
  2:       2       1
  3:       6       6       1
  4:      24      36      12      1
  5:     120     240     120     20      1
  6:     720    1800    1200    300     30     1
  7:    5040   15120   12600   4200    630    42    1
  8:   40320  141120  141120  58800  11760  1176   56  1
  9:  362880 1451520 1693440 846720 211680 28224 2016 72  1
  ...
Row n=10: [3628800, 16329600, 21772800, 12700800, 3810240, 635040, 60480, 3240, 90, 1]. - _Wolfdieter Lang_, Feb 01 2013
From _Peter Bala_, Feb 24 2025: (Start)
The array factorizes as an infinite product (read from right to left):
  /  1                \        /1             \^m /1           \^m /1           \^m
  |  2    1            |      | 0   1          |  |0  1         |  |1  1         |
  |  6    6   1        | = ...| 0   0   1      |  |0  1  1      |  |0  2  1      |
  | 24   36  12   1    |      | 0   0   1  1   |  |0  0  2  1   |  |0  0  3  1   |
  |120  240 120  20   1|      | 0   0   0  2  1|  |0  0  0  3  1|  |0  0  0  4  1|
  |...                 |      |...             |  |...          |  |...          |
where m = 2. Cf. A008277 (m = 1), A035342 (m = 3), A035469 (m = 4), A049029 (m = 5) A049385 (m = 6), A092082 (m = 7), A132056 (m = 8), A223511 - A223522 (m = 9 through 20), A001497 (m = -1), A004747 (m = -2), A000369 (m = -3), A011801 (m = -4), A013988 (m = -5). (End)

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Peter Bala, Factorising (r,b)-Stirling arrays
J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011, J. Int. Seq. 14 (2011) # 11.9.5
Jean-Paul Blaizot and Maciej A. Nowak, Large N_c confinement and turbulence, arXiv:0801.1859 [hep-th], 2008.
David Callan, Sets, Lists and Noncrossing Partitions, arXiv:0711.4841 [math.CO], 2007-2008.
Pietro Codara, Ottavio M. D'Antona, and Pavol Hell, A simple combinatorial interpretation of certain generalized Bell and Stirling numbers, arXiv preprint arXiv:1308.1700 [cs.DM], 2013.
Tom Copeland, Mathemagical Forests, Addendum to Mathemagical Forests, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions, A Class of Differential Operators and the Stirling Numbers
Siad Daboul, Jan Mangaldan, Michael Z. Spivey and Peter Taylor, The Lah Numbers and the n-th Derivative of exp(1/x), Math. Mag., 86 (2013), 39-47.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 33.
G. H. E. Duchamp et al., Feynman graphs and related Hopf algebras, J. Phys. (Conf Ser) 30 (2006) 107-118.
Rajesh Gopakumar and David J. Gross, Mastering the master field, arXiv:hep-th/9411021, 1994.
Gábor Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4. - From _Tom Copeland_, Oct 01 2015
Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.
Donald E. Knuth, Convolution polynomials, The Mathematica J., 2 (1992), 67-78.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From _N. J. A. Sloane_, Aug 21 2012
MacTutor History of Mathematics archive: Biography of Ivo Lah.
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Norihiro Nakashima and Shuhei Tsujie, Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species, arXiv:1904.09748 [math.CO], 2019. See p. 18.
Michael Penn, Lah Numbers and an appearance of exponential generating functions, YouTube video, 2025.
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019. See p. 4.
Tilman Piesk, Illustration of the first four rows
Kornelia Ufniarz and Grzegorz Siudem, Combinatorial origins of the canonical ensemble, arXiv:2008.00244 [math-ph], 2020. See p. 5.
Weiping Wang and Tianming Wang, Generalized Riordan arrays, Discrete Mathematics, Vol. 308, No. 24, 6466-6500.
Wikipedia, Lah number

Cf. A000262, A103194, A105220.
Triangle of Lah numbers (A008297) unsigned.
Cf. A111596 (signed triangle with extra n=0 row and m=0 column).
Cf. A130561 (for a natural refinement).
Cf. A094638 (for differential operator representation).
Cf. A248045 (central terms), A002868 (row maxima).
Cf, A059110.
Cf. A001263, A008277.
Cf. A089231 (triangle with mirrored rows).
Cf. A271703 (triangle with extra n=0 row and m=0 column).

Flat(List([1..10],n->List([1..n],k->Binomial(n,k)*Factorial(n-1)/Factorial(k-1)))); # Muniru A Asiru, Jul 25 2018

a105278 n k = a105278_tabl !! (n-1) !! (k-1)
a105278_row n = a105278_tabl !! (n-1)
a105278_tabl = [1] : f [1] 2 where
   f xs i = ys : f ys (i + 1) where
     ys = zipWith (+) ([0] ++ xs) (zipWith (*) [i, i + 1 ..] (xs ++ [0]))
-- Reinhard Zumkeller, Sep 30 2014, Mar 18 2013

/* As triangle */ [[Binomial(n,k)*Factorial(n-1)/Factorial(k-1): k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, Oct 31 2014

The triangle: for n from 1 to 13 do seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n) od;
the sequence: seq(seq(binomial(n,k)*(n-1)!/(k-1)!,k=1..n),n=1..13);
# The function BellMatrix is defined in A264428.
# Adds (1, 0, 0, 0, ...) as column 0.
BellMatrix(n -> (n+1)!, 9); # Peter Luschny, Jan 27 2016

nn = 9; a = x/(1 - x); f[list_] := Select[list, # > 0 &]; Flatten[Map[f, Drop[Range[0, nn]! CoefficientList[Series[Exp[y a], {x, 0, nn}], {x, y}], 1]]] (* Geoffrey Critzer, Dec 11 2011 *)
nn = 9; Flatten[Table[(j - k)! Binomial[j, k] Binomial[j - 1, k - 1], {j, nn}, {k, j}]] (* Jan Mangaldan, Mar 15 2013 *)
rows = 10;
t = Range[rows]!;
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 23 2018, after Peter Luschny *)
T[n_, n_] := 1; T[n_, k_] /;0Oliver Seipel, Dec 06 2024 *)

use ntheory ":all"; say join ", ", map { my $n=$; map { stirling($n,$,3) } 1..$n; } 1..9; # Dana Jacobsen, Mar 16 2017

T(n,k) = Sum_{m=n..k} |S1(n,m)|*S2(m,k), k>=n>=1, with Stirling triangles S2(n,m):=A048993 and S1(n,m):=A048994.
T(n,k) = C(n,k)*(n-1)!/(k-1)!.
Sum_{k=1..n} T(n,k) = A000262(n).
n*Sum_{k=1..n} T(n,k) = A103194(n) = Sum_{k=1..n} T(n,k)*k^2.
E.g.f. column k: (x^(k-1)/(1-x)^(k+1))/(k-1)!, k>=1.
Recurrence from Sheffer (here Jabotinsky) a-sequence [1,1,0,...] (see the W. Lang link under A006232): T(n,k)=(n/k)*T(n-1,m-1) + n*T(n-1,m). - Wolfdieter Lang, Jun 29 2007
The e.g.f. is, umbrally, exp[(.)!* L(.,-t,1)*x] = exp[t*x/(1-x)]/(1-x)^2 where L(n,t,1) = Sum_{k=0..n} T(n+1,k+1)*(-t)^k = Sum_{k=0..n} binomial(n+1,k+1)* (-t)^k / k! is the associated Laguerre polynomial of order 1. - Tom Copeland, Nov 17 2007
For this Lah triangle, the n-th row polynomial is given umbrally by
  n! C(B.(x)+1+n,n) = (-1)^n C(-B.(x)-2,n), where C(x,n)=x!/(n!(x-n)!),
  the binomial coefficient, and B_n(x)= exp(-x)(xd/dx)^n exp(x), the n-th Bell / Touchard / exponential polynomial (cf. A008277). E.g.,
  2! C(-B.(-x)-2,2) = (-B.(x)-2)(-B.(x)-3) = B_2(x) + 5*B_1(x) + 6 = 6 + 6x + x^2.
  n! C(B.(x)+1+n,n) = n! e^(-x) Sum_{j>=0} C(j+1+n,n)x^j/j! is a corresponding Dobinski relation. See the Copeland link for the relation to inverse Mellin transform. - Tom Copeland, Nov 21 2011
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)^2*d/dx. Cf. A008277 (D = (1+x)*d/dx), A035342 (D = (1+x)^3*d/dx), A035469 (D = (1+x)^4*d/dx) and A049029 (D = (1+x)^5*d/dx). - Peter Bala, Nov 25 2011
T(n,k) = Sum_{i=k..n} A130534(n-1,i-1)*A008277(i,k). - Reinhard Zumkeller, Mar 18 2013
Let E(x) = Sum_{n >= 0} x^n/(n!*(n+1)!). Then a generating function is exp(t)*E(x*t) = 1 + (2 + x)*t + (6 + 6*x + x^2)*t^2/(2!*3!) + (24 + 36*x + 12*x^2 + x^3)*t^3/(3!*4!) + ... . - Peter Bala, Aug 15 2013
P_n(x) = L_n(1+x) = n!*Lag_n(-(1+x);1), where P_n(x) are the row polynomials of A059110; L_n(x), the Lah polynomials of A105278; and Lag_n(x;1), the Laguerre polynomials of order 1. These relations follow from the relation between the iterated operator (x^2 D)^n and ((1+x)^2 D)^n with D = d/dx. - Tom Copeland, Jul 23 2018
Dividing each n-th diagonal by n!, where the main diagonal is n=1, generates the Narayana matrix A001263. - Tom Copeland, Sep 23 2020
T(n,k) = A089231(n,n-k). - Ron L.J. van den Burg, Dec 12 2021
T(n,k) = T(n-1,k-1) + (n+k-1)*T(n-1,k). - Bérénice Delcroix-Oger, Jun 25 2025

Stirling comments and e.g.f.s from Wolfdieter Lang, Apr 11 2007

A132440 Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 0

Offset: 0

Tom Copeland, Nov 13 2007, Nov 15 2007, Nov 22 2007, Dec 02 2007

Let M(t) = exp(t*T) = lim_{n->oo} (1 + t*T/n)^n.
Pascal matrix = [ binomial(n,k) ] = M(1) = exp(T), truncating the series gives the n X n submatrices.
Inverse Pascal matrix = M(-1) = exp(-T) = matrix for inverse binomial transform.
A(j) = T^j / j! equals the matrix [binomial(n,k) * delta(n-k-j)] where delta(n) = 1 if n=0 and vanishes otherwise (Kronecker delta); i.e., A(j) is a matrix with all the terms 0 except for the j-th lower (or main for j=0) diagonal, which equals that of the Pascal triangle. Hence the A(j)'s form a linearly independent basis for all matrices of the form [binomial(n,k) * d(n-k)] which include as a subset the invertible associated matrices of the list partition transform (LPT) of A133314.
For sequences with b(0) = 1, umbrally,
M[b(.)] = exp(b(.)*T) = [ binomial(n,k) * b(n-k) ] = matrices associated to b by LPT.
[M[b(.)]]^(-1) = exp(c(.)*T) = [ binomial(n,k) * c(n-k) ] = matrices associated to c, where c = LPT(b) . Or,
[M[b(.)]]^(-1) = exp[LPT(b(.))*T] = LPT[M(b(.))] = M[LPT(b(.))]= M[c(.)].
This is related to xDx, the iterated Laguerre transform and the general Euler transformation of a sequence through the comments in A132013 and A132014 and the relation [Sum_{k=0..n} binomial(n,k) * b(n-k) * d(k)] = M(b)*d, (n-th term). See also A132382.
If b(n,x) is a binomial type Sheffer sequence, then M[b(.,x)]*s(y) = s(x+y) when s(y) = (s(0,y),s(1,y),s(2,y),...) is an array for a Sheffer sequence with the same delta operator as b(n,x) and [M[b(.,x)]]^(-1) is given by the formulas above with b(n) replaced by b(n,x) as b(0,x)=1 for a binomial-type Sheffer sequence.
T = I - A132013 and conversely A132013 = I - T, which is the matrix representation for the iterated mixed order Laguerre transform characterized in A132013 (and A132014).
(I-T)^m generates the group [A132013]^m for m = 0,1,2,... discussed in A132014.
The inverse is 1/(I-T) = I + T + T^2 + T^3 + ... = [A132013]^(-1) = A094587 with the associated sequence (0!,1!,2!,3!,...) under the LPT.
And 1/(I-T)^2 = I + 2*T + 3*T^2 + 4*T^3 + ... = [A132013]^(-2) = A132159 with the associated sequence (1!,2!,3!,4!,...) under the LPT.
The matrix operation b = T*a can be characterized in several ways in terms of the coefficients a(n) and b(n), their o.g.f.'s A(x) and B(x), or e.g.f.'s EA(x) and EB(x).
1) b(0) = 0, b(n) = n * a(n-1),
2) B(x) = xDx A(x)
3) B(x) = x * Lag(1,-:xD:) A(x)
4) EB(x) = x * EA(x) where D is the derivative w.r.t. x, (:xD:)^j = x^j*D^j and Lag(n,x) is the Laguerre polynomial.
So the exponentiated operator can be characterized as
5) exp(t*T) A(x) = exp(t*xDx) A(x) = [Sum_{n=0,1,...} (t*x)^n * Lag(n,-:xD:)] A(x) = [exp{[t*u/(1-t*u)]*:xD:} / (1-t*u) ] A(x) (eval. at u=x) = A[x/(1-t*x)]/(1-t*x), a generalized Euler transformation for an o.g.f.,
6) exp(t*T) EA(x) = exp(t*x)*EA(x) = exp[(t+a(.))*x], gen. Euler trf. for an e.g.f.
7) exp(t*T) * a = M(t) * a = [Sum_{k=0..n} binomial(n,k) * t^(n-k) * a(k)].
The umbral extension of formulas 5, 6 and 7 gives formally
8) exp[c(.)*T] A(x) = exp(c(.)*xDx) A(x) = [Sum_{n>=0} (c(.)*x)^n * Lag(n,-:xD:)] A(x) = [exp{[c(.)*u/(1-c(.)*u)]*:xD:} / (1-c(.)*u) ] A(x) (eval. at u=x) = A[x/(1-c(.)*x)]/(1-c(.)*x), where the umbral evaluation should be applied only after a power series in c is obtained,
9) exp[c(.)*T] EA(x) = exp(c(.)*x)*EA(x) = exp[(c(.)+a(.))*x]
10) exp[c(.)*T] * a = M[c(.)] * a = [Sum_{k=0..n} binomial(n,k) * c(n-k) * a(k)] .
The n X n principal submatrix of T is nilpotent, in particular, [Tsub_n]^(n+1) = 0, n=0,1,2,3,....
Note (xDx)^n = x^n D^n x^n = x^n n! (:Dx:)^n/n! = x^n n! Lag(n,-:xD:).
The operator xDx is an important, classical operator explored by among others Dattoli, Al-Salam, Carlitz and Stokes and even earlier investigators.
For a recent treatment of xDx, DxD and more general operators see the paper "Laguerre-type derivatives: Dobinski relations and combinatorial identities". - Karol A. Penson, Sep 15 2009
See Copeland's link for generalized Laguerre functions and connection to fractional differ-integrals in exercises through (:Dx:)^a/a!=(D^a x^a)/a!. - Tom Copeland, Nov 17 2011
From Tom Copeland, Apr 25 2014: (Start)
Conjugation or "similarity" transformations of [dP]=A132440 have an operator interpretation (cf. A074909 and A238363):
In general, select two operators A and B such that A^n = F1(n,B) and B^n = F2(n,A); then A^n = F1(n,F2(.,A)) and B^n = F2(n,F1(.,B)), evaluated umbrally, i.e., F1(n,F2(.,x))=F2(n,F1(.,x))=x^n, implying the polynomials F1 and F2 are an umbral compositional inverse pair.
One such pair are the Bell polynomials Bell(n,x) and falling factorials (x)_n with Bell(n,:xD:)=(xD)^n and (xD)_n=:xD:^n (cf. A074909). Another are the Laguerre polynomials LN(n,x)= n!*Lag(n,x) (A021009), which are umbrally self-inverse, with LN(n,-:xD:)=:Dx:^n and LN(n,:Dx:)= (-:xD:)^n with :Dx:^n=D^n*x^n.
Evaluating, for n>=0, the operator derivative d(B^n)/dA = d(F2(n,A))/dA in the basis B^n, i.e., with A^n finally replaced by F1(n,B), or A^n=F1(.,B)^n=F1(n,B), is equivalent to the matrix conjugation
A)   [F2]*[dP]*[F1]
B) = [F2]*[dP]*[F2]^(-1)
C) = [F1]^(-1)*[dP]*[F1],
where [F1] is the lower triangular matrix with the n-th row the coefficients of F1(n,x) and analogously for [F2].
So, given 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).
D) dV(B)/dA = Rv * [F2]*[dP]*[F1] * Cv(B).
E) With A=D and B=D, F1(n,x)=F2(n,x)=x^n and [F1]=[F2]=I. Then d(B^n)/dA = d(D^n)/dD = n * D^(n-1); therefore, consistently [F2]*[dP]*[F1] = [dP] and dV(D)/dD = Rv * [dP] * Cv(D). (End)

			Matrix T begins
  0;
  1,0;
  0,2,0;
  0,0,3,0;
  0,0,0,4,0;
  ...

T. Mansour and M. Schork, Commutation Relations, Normal Ordering, and Stirling Numbers, Chapman and Hall/CRC, 2015, (x^n D^n x^n on p. 187).

Robert Israel, Table of n, a(n) for n = 0..10000
W. A. Al-Salam, Operational representations for the Laguerre and other polynomials, Duke Math. Jour., vol. 31 (1964), pp. 127-142.
Tom Copeland, Fractional Calculus, Gamma Classes, the Riemann Zeta Function, and an Appell Pair of Sequences.
Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator.
Tom Copeland, Infinigens, the Pascal Pyramid, and the Witt and Virasoro Algebras.
Tom Copeland, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions (pdf).
Tom Copeland, Mathemagical Forests.
Tom Copeland, Mellin Interpolation of Differential Ops and Associated Infinigens and Appell Polynomials: The Ordered, Laguerre, and Scherk-Witt-Lie Diff Ops.
G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009. (Cf. Viennot's Laguerre histoires.)
K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, arXiv:0904.0369 [math-ph], 2009.
K. A. Penson, P. Blasiak, A. Horzela, G.H.E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, Journal of Mathematical Physics vol. 50, (2009) 083512.
G. Stokes, Note on certain formulae in the calculus of operations, Proceedings of the Royal Society of Edinburgh, IX, pp. 101-102, 1876.

seq(op([0$i,i]),i=1..20); # Robert Israel, Oct 02 2015

Table[PadLeft[{n, 0}, n+1], {n, 0, 11}] // Flatten (* Jean-François Alcover, Apr 30 2014 *)

T = log(P) with the Pascal matrix P:=A007318. This should be read as T_N = log(P_N) with P_N the N X N matrix P, N>=2. Because P_N is lower triangular with all diagonal elements 1, the series log(1_N-(1_N-P_N)) stops after N-1 terms because (1_N-P_N)^N is the 0_N-matrix. - Wolfdieter Lang, Oct 14 2010
Given a polynomial sequence p_n(x) with p_0(x)=1 and the lowering and raising operators L and R defined by L p_n(x) = n * p_(n-1)(x) and R p_n(x) = p_(n+1)(x), the matrix T represents the action of R*L*R in the p_n(x) basis. For p_n(x) = x^n, L = D = d/dx and R = x. For p_n(x) = x^n/n!, L = DxD and R = D^(-1). - Tom Copeland, Oct 25 2012
From Tom Copeland, Apr 26 2014: (Start)
A) T = exp(A238385-I) - I
B)   = [St1]*P*[St2] - I
C)   = [St1]*P*[St1]^(-1) - I
D)   = [St2]^(-1)*P*[St2] - I
E)   = [St2]^(-1)*P*[St1]^(-1) - I
where P=A007318, [St1]=padded A008275 just as [St2]=A048993=padded A008277, and I=identity matrix. (End)
From Robert Israel, Oct 02 2015: (Start)
G.f. Sum_{k >= 1} k x^((k+3/2)^2/2 - 17/8) is related to Jacobi theta functions.
If 8*n+17 = y^2 is a square, then a(n) = (y-3)/2, otherwise a(n) = 0. (End)

Missing zero added in table by Tom Copeland, Feb 25 2014

A260832 a(n) = numerator(Jtilde2(n)).

Original entry on oeis.org

1, 3, 41, 147, 8649, 32307, 487889, 1856307, 454689481, 1748274987, 26989009929, 104482114467, 6488426222001, 25239009088827, 393449178700161, 1535897056631667, 1537112996582116041, 6016831929058214523, 94316599529950360769, 369994845516850143483, 23244865440911268112681

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde2(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(2), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.

G. C. Greubel, Table of n, a(n) for n = 0..830
Takashi Ichinose and Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto and Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 1).
Ling Long, Robert Osburn and Holly Swisher, On a conjecture of Kimoto and Wakayama, Proc. Amer. Math. Soc. 144 (2016), 4319-4327.

Cf. A056982 (denominators), A013661 (zeta(2)), A264541 (Jtilde3).

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a := n -> numer(simplify(hypergeom([1/2, 1/2, -n], [1, 1], 1))):
seq(a(n), n = 0..20); # Peter Luschny, Dec 08 2022

Numerator[Table[Sum[ (-1)^k*Binomial[-1/2, k]^2*Binomial[n, k], {k, 0, n}], {n,0,50}]] (* G. C. Greubel, Feb 15 2017 *)

a(n) = numerator(sum(k=0, n, (-1)^k*binomial(-1/2,k)^2*binomial(n, k)));

a(n) = numerator(sum(k=0, n, binomial(2*k, k)*binomial(4*k, 2*k)* binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))) / (2^(4*n)* binomial(2*n,n)));

Jtilde2(n) = J2(n)/J2(0) with J2(0) = 3*zeta(2) (normalization).
And 4n^2*J2(n) - (8n^2-8n+3)*J2(n-1) + 4(n-1)^2*J2(n-2) = 0 with J2(0) = 3*zeta(2) and J2(1) = 9*zeta(2)/4.
Jtilde2(n) = Sum_{k=0..n} (-1)^k*binomial(-1/2,k)^2*binomial(n,k).
Jtilde2(n) = Sum_{k=0..n} binomial(2*k,k)*binomial(4*k,2*k)*binomial(2*(n-k),n-k)*binomial(4*(n-k),2*(n-k))/(2^(4*n)*binomial(2*n,n)).
From Andrey Zabolotskiy, Oct 04 2016 and Dec 08 2022: (Start)
Jtilde2(n) = Integral_{ x >= 0 } (L_n(x))^2*exp(-x)/sqrt(Pi*x) dx, where L_n(x) is the Laguerre polynomial (A021009).
G.f. of Jtilde2(n): 2F1(1/2,1/2;1;z/(z-1))/(1-z).
Jtilde2(n) = A143583(n) / 16^n. (End)
a(n) = numerator(hypergeom([1/2, 1/2, -n], [1, 1], 1)). - Peter Luschny, Dec 08 2022

A009940 a(n) = n!*L_{n}(1), where L_{n}(x) is the n-th Laguerre polynomial.

Original entry on oeis.org

1, 0, -1, -4, -15, -56, -185, -204, 6209, 112400, 1520271, 19165420, 237686449, 2944654296, 36392001815, 441823808804, 5066513855745, 49021548330016, 202510138910239, -8592616658156580, -399625593156546319, -12547977062999220760, -350780150063930345241, -9361105704321312342764

Offset: 0

Wouter Meeussen

Previous name was: Form the iterate f[ f[ .. f[ 1 ] ] ] or f^n [ 1 ] with f[ stuff ] defined as ( stuff - Integrate[ stuff over x ] ), set x=1 and multiply by n!.

This presumably means the recurrence L(n+1,x) = L(n,x) - Integral_{t=0..x} L(n,t) dt with L(0,x) = 1, which is satisfied by the Laguerre polynomials. - Robert Israel, Jan 09 2015

			The first few f[ x ] are 1, 1 - x, 1 - 2*x + x^2/2, 1 - 3*x + (3*x^2)/2 - x^3/6, giving the values 1, 0, -1/2, -2/3, ...

G. C. Greubel, Table of n, a(n) for n = 0..449
Anne-Maria Ernvall-Hytönen and Tapani Matala-aho, Explicit estimates for the sum Sum_{k=0..n} k!*binomial(n,k)^2*(-1)^k, arXiv:2310.11468 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Laguerre Polynomial.

Row sums of A021009.

I:=[1,0]; [n le 2 select I[n] else 2*(n-1)*Self(n-1) - (n-1)^2*Self(n-2): n in [1..30]]; // G. C. Greubel, Feb 05 2018

seq(coeff(series(exp(x/(x-1))/(1-x),x,50),x,i)*i!, i=0..20);
A009940:=proc(n) options remember: if n<2 then RETURN([1,0][n+1]) else RETURN(2*(n-1)*A009940(n-1)-(n-1)^2*A009940(n-2)) fi: end; seq(A009940(n), n=0..20);
with(orthopoly): seq(n!*L(n,1), n=0..20); # C. Ronaldo, Dec 19 2004

(NestList[ #-Integrate[ #, x ]&, 1, 32 ]/.x:>1) Range[ 0, 32 ]!
Table[ n! LaguerreL[ n, 1 ], {n, 18} ]
Table[n! Sum[(-1)^k Binomial[n, k]/k!, {k, 0, n}], {n, 0, 10}] (* Benedict W. J. Irwin, Apr 20 2017 *)

my(x='x+O('x^30)); Vec(serlaplace(exp(x/(x-1))/(1-x))) \\ G. C. Greubel, Feb 05 2018

a(n) = n!*pollaguerre(n, 0, 1); \\ Michel Marcus, Feb 06 2021

From C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004: (Start)
E.g.f.: exp(x/(x-1))/(1-x).
D-finite with recurrence a(n) = 2*(n-1)*a(n-1)-(n-1)^2*a(n-2) for n>1, a(0)=1, a(1)=0.
a(n) = n!*Laguerre(n, 1). (End)
a(n) = n!*Sum_{k=0..n} (-1)^k*Binomial(n,k)/k!. - Benedict W. J. Irwin, Apr 20 2017
a(n) ~ sqrt(2) * n^(n + 1/4) / exp(n - 1/2) * (sin(2*sqrt(n) + Pi/4) + (17*cos(2*sqrt(n) + Pi/4)) / (48*sqrt(n))). - Vaclav Kotesovec, Feb 25 2019
Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselJ(0,2*sqrt(x)). - Ilya Gutkovskiy, Jul 17 2020
From Peter Bala, Mar 12 2023: (Start)
a(n) = n! * [x^n] (1 + x)^n*exp(-x).
a(n + k) == (-1)^k*a(n) (mod k) for all n and k. It follows that the sequence a(n) taken modulo 2*k is periodic with the period dividing 2*k. See A047974. (End)
a(n) = Sum_{k=0..n} (-1)^(n-k)*k!*binomial(n,k)^2. - Ridouane Oudra, Jul 03 2025
a(n) = (-1)^n*U(-n, 1, 1)  where U(a, b, c) is the confluent hypergeometric function. - Stefano Spezia, Jul 04 2025

New name using a formula from W. Meeussen's program by Peter Luschny, Jan 09 2015

A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 9, 21, 14, 0, 1, 14, 56, 84, 42, 0, 1, 20, 120, 300, 330, 132, 0, 1, 27, 225, 825, 1485, 1287, 429, 0, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 0, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 0, 1, 54, 936, 7644, 34398, 91728

Offset: 0

Philippe Deléham, Aug 05 2003

Mirror image of triangle A133336. - Philippe Deléham, Dec 10 2008
From Tom Copeland, Oct 09 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = t
P(3,t) = t + 2 t^2
P(4,t) = t + 5 t^2 +  5 t^3
P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4
The o.g.f. A(x,t) = {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)] (see Drake et al.).
B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.
Let h(x,t) = 1/(dB/dx) = (1-x)^2/(1+(1+t)*x*(x-2)) = 1/(1-t(2x+3x^2+4x^3+...)), an o.g.f. in x for row polynomials in t of A181289. Then P(n,t) is given by (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A = exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). These results are a special case of A133437 with u(x,t) = B(x,t), i.e., u_1=1 and (u_n)=-t for n > 1. See A001003 for t = 1. (End)
Let U(x,t) = [A(x,t)-x]/t, then U(x,0) = -dB(x,t)/dt and U satisfies dU/dt = UdU/dx, the inviscid Burgers' equation (Wikipedia), also called the Hopf equation (see Buchstaber et al.). Also U(x,t) = U(A(x,t),0) = U(x+tU,0) since U(x,0) = [x-B(x,t)]/t. - Tom Copeland, Mar 12 2012
Diagonals of A132081 are essentially rows of this sequence. - Tom Copeland, May 08 2012
T(r, s) is the number of [0,r]-covering hierarchies with s segments (see Kreweras). - Michel Marcus, Nov 22 2014
From Yu Hin Au, Dec 07 2019: (Start)
T(n,k) is the number of small Schröder n-paths (lattice paths from (0,0) to (2n,0) using steps U=(1,1), F=(2,0), D=(1,-1) with no F step on the x-axis) that has exactly k U steps.
T(n,k) is the number of Schröder trees (plane rooted tree where each internal node has at least two children) with exactly n+1 leaves and k internal nodes. (End)

			Triangle starts:
  1;
  0,  1;
  0,  1,  2;
  0,  1,  5,  5;
  0,  1,  9, 21, 14;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
V. Buchstaber and E. Bunkova,Elliptic formal group laws, integral Hirzebruch genera and Kirchever genera,, arXiv:1010.0944 [math-ph], 2010 (see p. 19).
V. Buchstaber and T. Panov, Toric Topology. Chapter 1: Geometry and Combinatorics of Polytopes,, arXiv:1102.1079 [math.CO], 2011-2012 (see p. 41).
G. Chatel, V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
T. Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra, 2014.
T. Copeland, Lagrange a la Lah, 2011.
B. Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973, p. 21-22.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Diagonals: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281.
The diagonals (except for A000007) are also the diagonals of A033282.
Row sums: A001003 (Schroeder numbers).
Cf. A033282, A084938.
Cf. A001003, A008297, A021009, A132081, A133437, A181289.

Table[Boole[n == 2] + If[# == -1, 0, Binomial[n - 3, #] Binomial[n + # - 1, #]/(# + 1)] &[k - 1], {n, 2, 12}, {k, 0, n - 2}] // Flatten (* after Jean-François Alcover at A033282, or *)
Table[If[n == 0, 1, Binomial[n, k] Binomial[n + k, k - 1]/n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)

t(n, k) = if (n==0, 1, binomial(n, k)*binomial(n+k, k-1)/n);
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n,k), ", ");); print(););} \\ Michel Marcus, Nov 22 2014

Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
For k>0, T(n, k) = binomial(n-1, k-1)*binomial(n+k, k)/(n+1); T(0, 0) = 1 and T(n, 0) = 0 if n > 0. [corrected by Marko Riedel, May 04 2023]
Sum_{k>=0} T(n, k)*2^k = A107841(n). - Philippe Deléham, May 26 2005
Sum_{k>=0} T(n-k, k) = A005043(n). - Philippe Deléham, May 30 2005
T(n, k) = A108263(n+k, k). - Philippe Deléham, May 30 2005
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*5^k*(-2)^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A154825(n). - Philippe Deléham, Jan 17 2009
Umbrally, P(n,t) = Lah[n-1,-t*a.]/n! = (1/n)*Sum_{k=1..n-1} binomial(n-2,k-1)a_k t^k/k!, where (a.)^k = a_k = (n-1+k)!/(n-1)!, the rising factorial, and Lah(n,t) = n!*Laguerre(n,-1,t) are the Lah polynomials A008297 related to the Laguerre polynomials of order -1. - Tom Copeland, Oct 04 2014
T(n, k) = binomial(n, k)*binomial(n+k, k-1)/n, for k >= 0; T(0, 0) = 1 (see Kreweras, p. 21). - Michel Marcus, Nov 22 2014
P(n,t) = Lah[n-1,-:Dt:]/n! t^(n-1) with (:Dt:)^k = (d/dt)^k t^k = k! Laguerre(k,0,-:tD:) with (:tD:)^j = t^j D^j. The normalized Laguerre polynomials of 0 order are given in A021009. - Tom Copeland, Aug 22 2016

Typo in a(60) corrected by Michael De Vlieger, Nov 21 2019

A001809 a(n) = n! * n(n-1)/4.

Original entry on oeis.org

0, 0, 1, 9, 72, 600, 5400, 52920, 564480, 6531840, 81648000, 1097712000, 15807052800, 242853811200, 3966612249600, 68652904320000, 1255367393280000, 24186745110528000, 489781588488192000, 10400656084955136000, 231125690776780800000, 5364548928029491200000

Offset: 0

N. J. A. Sloane

a(n) = n!*n*(n-1)/4 gives the total number of inversions in all the permutations of [n]. [Stern, Terquem] Proof: For fixed i,j and for fixed I,J (i < j, I > J, 1 <= i,j,I,J <= n), we have (n-2)! permutations p of [n] for which p(i)=I and p(j)=J (permute {1,2,...,n} \ {I,J} in the positions (1,2,...,n) \ {i,j}). There are n*(n-1)/2 choices for the pair (i,j) with i < j and n*(n-1)/2 choices for the pair (I,J) with I > J. Consequently, the total number of inversions in all the permutations of [n] is (n-2)!*(n*(n-1)/2)^2 = n!*n*(n-1)/4. - Emeric Deutsch, Oct 05 2006
To state this another way, a(n) is the number of occurrences of the pattern 12 in all permutations of [n]. - N. J. A. Sloane, Apr 12 2014
Equivalently, this is the total Denert index of all permutations on n letters (cf. A008302). - N. J. A. Sloane, Jan 20 2014
Also coefficients of Laguerre polynomials. a(n)=A021009(n,2), n >= 2.
a(n) is the number of edges in the Cayley graph of the symmetric group S_n with respect to the generating set consisting of transpositions. - Avi Peretz (njk(AT)netvision.net.il), Feb 20 2001
a(n+1) is the sum of the moments over all permutations of n. E.g. a(4) is [1,2,3].[1,2,3] + [1,3,2].[1,2,3] + [2,1,3].[1,2,3] + [2,3,1].[1,2,3] + [3,1,2].[1,2,3] + [3,2,1].[1,2.3] = 14 + 13 + 13 + 11 + 11 + 10 = 72. - Jon Perry, Feb 20 2004
Derivative of the q-factorial [n]!, evaluated at q=1. Example: a(3)=9 because (d/dq)[3]!=(d/dq)((1+q)(1+q+q^2))=2+4q+3q^2 is equal to 9 at q=1. - Emeric Deutsch, Apr 19 2007
Also the number of maximal cliques in the n-transposition graph for n > 1. - Eric W. Weisstein, Dec 01 2017
a(n-1) is the number of trees on [n], rooted at 1, with exactly two leaves.  A leaf is a non-root vertex of degree 1. - Nikos Apostolakis, Dec 27 2021

			G.f. = x^2 + 9*x^3 + 72*x^4 + 600*x^5 + 5400*x^6 + 52920*x^7 + ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 799.
Simon Altmann and Eduardo L. Ortiz, Editors, Mathematical and Social Utopias in France: Olinde Rodrigues and His Times, Amer. Math. Soc., 2005.
David M. Bressoud, Proofs and Confirmations, Camb. Univ. Press, 1999; p. 90.
Cornelius Lanczos, Applied Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 519.
Edward M. Reingold, Jurg Nievergelt and Narsingh Deo, Combinatorial Algorithms, Prentice-Hall, 1977, section 7.1, p. 287.
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).
Olry Terquem, Liouville's Journal, 1838.

T. D. Noe, Table of n, a(n) for n=0..100
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].
Eric Babson and Einar Steingrimsson, Generalized Permutation Patterns and Classification of the Mahonian Statistics, Séminaire Lotharingien de Combinatoire, B44b (2000), 18 pp.
Geoffrey Critzer, Combinatorics of Vector Spaces over Finite Fields, Master's thesis, Emporia State University, 2018.
Karl Dienger, Beiträge zur Lehre von den arithmetischen und geometrischen Reihen höherer Ordnung, Jahres-Bericht Ludwig-Wilhelm-Gymnasium Rastatt, Rastatt, 1910. [Annotated scanned copy]
FindStat - Combinatorial Statistic Finder, The Denert index of a permutation
Dominique Foata and Marcel-Paul Schützenberger, Major Index and inversion number of permutations , Math. Nachr. 83 (1978), 143-159
Dexter Jane L. Indong and Gilbert R. Peralta, Inversions of permutations in Symmetric, Alternating, and Dihedral Groups, JIS, Vol. 11 (2008), Article 08.4.3.
Warren P. Johnson, Review of Altmann-Ortiz book, Amer. Math. Monthly, Vol. 114, No. 8 (2007), pp. 752-758.
Cornelius Lanczos, Applied Analysis. (Annotated scans of selected pages)
J. Ser, Les Calculs Formels des Séries de Factorielles, Gauthier-Villars, Paris, 1933 [Local copy].
J. Ser, Les Calculs Formels des Séries de Factorielles. (Annotated scans of some selected pages)
M. Stern, Aufgaben, Journal für die reine und angewandte Mathematik, Vol. 18 (1838), p. 100.
Thotsaporn Thanatipanonda, Inversions and major index for permutations, Math. Mag., Vol. 77, No. 2 (April 2004), pp. 136-140.
Eric Weisstein's World of Mathematics, Maximal Clique.
Eric Weisstein's World of Mathematics, Transposition Graph.
Index entries for sequences related to Laguerre polynomials.

Cf. A034968 (the inversion numbers of permutations listed in alphabetic order). See also A053495 and A064038.

Cf. A001620, A008302, A099285.

[Factorial(n)*n*(n-1)/4: n in [0..20]]; // Vincenzo Librandi, Jun 15 2015

A001809 := n->n!*n*(n-1)/4;
with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=1)}, labeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m), m=0..19); # Zerinvary Lajos, Feb 07 2008
with (combstruct):with (combinat):a:=proc(m) [ZL, {ZL=Set(Cycle(Z, card>=m))}, labeled]; end: ZLL:=a(1):seq(count(ZLL, size=n)*binomial(n,2)/2, n=0..21); # Zerinvary Lajos, Jun 11 2008

Table[n! n (n - 1)/4, {n, 0, 18}]
Table[n! Binomial[n, 2]/2, {n, 0, 20}] (* Eric W. Weisstein, Dec 01 2017 *)
Coefficient[Table[n! LaguerreL[n, x], {n, 20}], x, 2] (* Eric W. Weisstein, Dec 01 2017 *)

PARI
```
{a(n) = n! * n * (n-1) / 4};
```

[factorial(m) * binomial(m, 2) / 2 for m in range(19)]  # Zerinvary Lajos, Jul 05 2008

E.g.f.: (1/2)*x^2/(1-x)^3.
a(n) = a(n-1)*n^2/(n-2), n > 2; a(2)=1.
a(n) = n*a(n-1) + (n-1)!*n*(n-1)/2, a(1) = 0, a(2) = 1; a(n) = sum (first n! terms of A034968); a(n) = sum of the rises j of permutations (p(j)If we define f(n,i,x) = Sum_{k=i..n} (Sum_{j=i..k}(C(k,j)*Stirling1(n,k)*Stirling2(j,i)*x^(k-j))) then a(n)=(-1)^n*f(n,2,-3), (n>=2). - Milan Janjic, Mar 01 2009
a(n) = Sum_k k*A008302(n,k). - N. J. A. Sloane, Jan 20 2014
a(n+2) = n*n!*(n+1)^2 / 4 = A000142(n) * (A000292(n) + A000330(n))/2 = sum of the cumulative sums of all the permutations of numbers from 1 to n, where A000142(n) = n! and sequences A000292(n) and A000330(n) are sequences of minimal and maximal values of cumulative sums of all the permutations of numbers from 1 to n. - Jaroslav Krizek, Sep 13 2014
From Amiram Eldar, Feb 15 2022: (Start)
Sum_{n>=2} 1/a(n) = 12 - 4*e.
Sum_{n>=2} (-1)^n/a(n) = 8*gamma - 4 - 4/e - 8*Ei(-1), where gamma is Euler's constant (A001620) and -Ei(-1) is the negated exponential integral at -1 (A099285). (End)

More terms and new description from Michael Somos, May 19 2000

Simpler description from Emeric Deutsch, Oct 05 2006

A066667 Coefficient triangle of generalized Laguerre polynomials (a=1).

Original entry on oeis.org

1, 2, -1, 6, -6, 1, 24, -36, 12, -1, 120, -240, 120, -20, 1, 720, -1800, 1200, -300, 30, -1, 5040, -15120, 12600, -4200, 630, -42, 1, 40320, -141120, 141120, -58800, 11760, -1176, 56, -1, 362880, -1451520, 1693440, -846720, 211680, -28224, 2016

Offset: 0

Christian G. Bower, Dec 17 2001

Same as A008297 (Lah triangle) except for signs.
Row sums give A066668. Unsigned row sums give A000262.
The Laguerre polynomials L(n;x;a=1) under discussion are connected with Hermite-Bell polynomials p(n;x) for power -1 (see also A215216) defined by the following relation: p(n;x) := x^(2n)*exp(x^(-1))*(d^n exp(-x^(-1))/dx^n). We have L(n;x;a=1)=(-x)^(n-1)*p(n;1/x), p(n+1;x)=x^2(dp(n;x)/dx)+(1-2*n*x)p(n;x), and p(1;x)=1, p(2;x)=1-2*x, p(3;x)=1-6*x+6*x^2, p(4;x)=1-12*x+36*x^2-24*x^3, p(5;x)=1-20*x+120*x^2-240*x^3+120*x^4. Note that if we set w(n;x):=x^(2n)*p(n;1/x) then w(n+1;x)=(w(n;x)-(dw(n;x)/dx))*x^2, which is almost analogous to the recurrence formula for Bell polynomials B(n+1;x)=(B(n;x)+(dB(n;x)/dx))*x. - Roman Witula, Aug 06 2012.

			Triangle a(n,m) begins
n\m     0        1       2       3      4      5    6   7  8
0:      1
1:      2       -1
2:      6       -6       1
3:     24      -36      12      -1
4:    120     -240     120     -20      1
5:    720    -1800    1200    -300     30     -1
6:   5040   -15120   12600   -4200    630    -42    1
7:  40320  -141120  141120  -58800  11760  -1176   56  -1
8: 362880 -1451520 1693440 -846720 211680 -28224 2016 -72  1
9: 3628800, -16329600, 21772800, -12700800, 3810240, -635040, 60480, -3240, 90, -1.
Reformatted and extended by _Wolfdieter Lang_, Jan 31 2013.
From _Wolfdieter Lang_, Jan 31 2013 (Start)
Recurrence (standard): a(4,2) = 2*4*12 - (-36) - 4*3*1 = 120.
Recurrence (simple): a(4,2) = 7*12 - (-36) = 120. (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 778 (22.5.17).
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 95 (4.1.62)
R. Witula, E. Hetmaniok, and D. Slota, The Hermite-Bell polynomials for negative powers, (submitted, 2012)

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 >= n >= 150, flattened).
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].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 778 (22.5.17).
Mathias Pétréolle and Alan D. Sokal, Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions, arXiv:1907.02645 [math.CO], 2019.
Jian Zhou, On Some Mathematics Related to the Interpolating Statistics, arXiv:2108.10514 [math-ph], 2021.

Cf. A021009, A062137, A062138, A062139, A062140, A089231.

A066667 := (n, k) -> (-1)^k*binomial(n, k)*(n + 1)!/(k + 1)!:
for n from 0 to 9 do seq(A066667(n,k), k = 0..n) od; # Peter Luschny, Jun 22 2022

Table[(-1)^m*Binomial[n, m]*(n + 1)!/(m + 1)!, {n, 0, 8}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 04 2019 *)

row(n) = Vecrev(n!*pollaguerre(n, 1)); \\ Michel Marcus, Feb 06 2021

E.g.f. (relative to x, keep y fixed): A(x)=(1/(1-x))^2*exp(x*y/(x-1)).
From Wolfdieter Lang, Jan 31 2013: (Start)
a(n,m) = (-1)^m*binomial(n,m)*(n+1)!/(m+1)!, n >= m >= 0. [corrected by Georg Fischer, Oct 26 2022]
Recurrence from standard three term recurrence for orthogonal generalized Laguerre polynomials {P(n,x):=n!*L(1,n,x)}:
  P(n,x) = (2*n-x)*P(n-1,x) - n*(n-1)*P(n-2), n>=1, P(-1,x) = 0, P(0,x) = 1.
  a(n,m) = 2*n*a(n-1,m) - a(n-1,m-1) - n*(n-1)*a(n-2,m), n>=1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < m.
Simplified recurrence from explicit form of a(n,m):
  a(n,m) = (n+m+1)*a(n-1,m) - a(n-1,m-1), n >= 1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < m.
(End)

A062139 Coefficient triangle of generalized Laguerre polynomials n!*L(n,2,x) (rising powers of x).

Original entry on oeis.org

1, 3, -1, 12, -8, 1, 60, -60, 15, -1, 360, -480, 180, -24, 1, 2520, -4200, 2100, -420, 35, -1, 20160, -40320, 25200, -6720, 840, -48, 1, 181440, -423360, 317520, -105840, 17640, -1512, 63, -1, 1814400, -4838400, 4233600

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,2,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^3. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = Sum_{k=0..n} binomial(n,k)*s(k,x)*p(n-k,y), with polynomials p(n,x) = Sum_{m=1..n} |A008297(n,m)|*(-x)^m, n >= 1 and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).
This unsigned matrix is embedded in the matrix for n!*L(n,-2,-x). Introduce 0,0 to each unsigned row and then add 1,-1,1 to the array as the first two rows to generate n!*L(n,-2,-x). - Tom Copeland, Apr 20 2014
The unsigned n-th row reverse polynomial equals the numerator polynomial of the finite continued fraction 1 - x/(1 + (n+1)*x/(1 + n*x/(1 + n*x/(1 + ... + 2*x/(1 + 2*x/(1 + x/(1 + x/(1)))))))). Cf. A089231. The denominator polynomial of the continued fraction is the (n+1)-th row polynomial of A144084. An example is given below. - Peter Bala, Oct 06 2019

			Triangle begins:
     1;
     3,    -1;
    12,    -8,    1;
    60,   -60,   15,   -1;
   360,  -480,  180,  -24,  1;
  2520, -4200, 2100, -420, 35, -1;
  ...
2!*L(2,2,x) = 12 - 8*x + x^2.
Unsigned row 3 polynomial in reverse form as the numerator of a continued fraction: 1 - x/(1 + 4*x/(1 + 3*x/(1 + 3*x/(1 + 2*x/(1 + 2*x/(1 + x/(1 + x))))))) = (60*x^3 + 60*x^2 + 15*x + 1)/(24*x^4 + 96*x^3 + 72*x^2 + 16*x + 1). - _Peter Bala_, Oct 06 2019

Indranil Ghosh, Rows 0..125, flattened
Robert S. Maier, Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers, arXiv:2308.10332 [math.CO], 2023. See p. 19.
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001710, A005990, A005461, A062193-A062195. The row sums (signed) give A062197, the row sums (unsigned) give A052852.

Cf. A021009, A062137-A062140, A066667, A089231, A144084.

with(PolynomialTools):
p := n -> (n+2)!*hypergeom([-n],[3],x)/2:
seq(CoefficientList(simplify(p(n)), x), n=0..9); # Peter Luschny, Apr 08 2015

Flatten[Table[((-1)^m)*n!*Binomial[n+2,n-m]/m!,{n,0,8},{m,0,n}]] (* Indranil Ghosh, Feb 24 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1(((-1)^k)*n!*binomial(n+2, n-k)/k!, ", ");); print(););} \\ Michel Marcus, May 06 2014

row(n) = Vecrev(n!*pollaguerre(n, 2)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n,r):return f(n)//f(r)//f(n-r)
i=0
for n in range(16):
    for m in range(n+1):
        i += 1
        print(i,((-1)**m)*f(n)*C(n+2,n-m)//f(m)) # Indranil Ghosh, Feb 24 2017

from functools import cache
@cache
def T(n, k):
    if k < 0 or k > n: return 0
    if k == n: return (-1)**n
    return (n + k + 2) * T(n-1, k) - T(n-1, k-1)
for n in range(7): print([T(n,k) for k in range(n + 1)])
# Peter Luschny, Mar 25 2024

T(n, m) = ((-1)^m)*n!*binomial(n+2, n-m)/m!.
E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^3), m >= 0.
n!*L(n,2,x) = (n+2)!*hypergeom([-n],[3],x)/2. - Peter Luschny, Apr 08 2015
From Werner Schulte, Mar 24 2024: (Start)
T(n, k) = (n+k+2) * T(n-1, k) - T(n-1, k-1) with initial values T(0, 0) = 1 and T(i, j) = 0 if j < 0 or j > i.
T = T^(-1), i.e., T is matrix inverse of T. (End)

A084950 Array of coefficients of denominator polynomials of the n-th approximation of the continued fraction x/(1+x/(2+x/(3+..., related to Laguerre polynomial coefficients.

Original entry on oeis.org

1, 1, 2, 1, 6, 4, 24, 18, 1, 120, 96, 9, 720, 600, 72, 1, 5040, 4320, 600, 16, 40320, 35280, 5400, 200, 1, 362880, 322560, 52920, 2400, 25, 3628800, 3265920, 564480, 29400, 450, 1, 39916800, 36288000, 6531840, 376320, 7350, 36, 479001600, 439084800, 81648000, 5080320, 117600, 882, 1

Offset: 0

Gary W. Adamson, Jun 14 2003

A factorial triangle, with row sums A001040(n+1), n >= 0.
Conjecture: also coefficient triangle of the denominators of the (n-th convergents to) the continued fraction w/(1+w/(2+w/(3+w/... This continued fraction converges to 0.697774657964... = BesselI(1,2)/BesselI(0,2) for w=1. For instance, the denominator of w/(1 + w/(2 + w/(3 + w/(4 + w/5)))) equals 120 + 96*w + 9*w^2. - Wouter Meeussen, Aug 08 2010
For general w, Bill Gosper showed it equals n!*2F3([1/2-n/2,-n/2], [1,-n,-n], 4*w). - Wouter Meeussen, Jan 05 2013
From Wolfdieter Lang, Mar 02 2013: (Start)
The row length sequence of this array is 1 + floor(n/2) = A008619(n), n >= 0.
The continued fraction 0 + K_{k>=1}(x/k) = x/(1+x/(2+x/(3+... has n-th approximation P(n,x)/Q(n,x). These polynomials satisfy the recurrence q(n,x) = n*q(n-1,x) + x*q(n-2,x), for q replaced by P or Q with inputs P(-1,x) = 1, P(0,x) = 0 and Q(-1,x) = 0 and Q(0,1) = 1. The present array provides the Q-coefficients: Q(n,x) = sum(a(n,m)*x^m, m=0 .. floor(n/2)), n >= 0. For the P(n,x)/x coefficients see the companion array A221913. This proves the first part of W. Meeussen's conjecture given above.
The solution with input q(-1,x) = a and q(0,x) = b is then, due to linearity, q(a,b;n,x) = a*P(n,x) + b*Q(n,x).  The motivation to look at the q(n,x) recurrence came from an e-mails from Gary Detlefs, who considered integer x and various inputs and gave explicit formulas.
This array coincides with the SW-NE diagonals of the unsigned Laguerre polynomial coefficient triangle |A021009|.
The entries a(n,m) have a combinatorial interpretation in terms of certain so-called labeled Morse code polynomials using dots (length 1) and dashes (of length 2). a(n,m) is the number of possibilities to decorate the n positions 1,2,...,n with m dashes, m from {0, 1, ..., floor(n/2)}, and n-2*m dots. A dot at position k has a weight k and each dash between two neighboring positions has a label x. a(n,m) is the sum of these labeled Morse codes with m dashes after the label x^m has been divided out. E.g., a(5,2) = 5 + 3 + 1 = 9 from the 3 codes: dash dash dot, dash dot dash, and dot dash dash, or (12)(34)5, (12)3(45) and 1(23)(45) with labels (which are in general multiplicative) 5*x^2, 3*x^2 and 1*x^2 , respectively. For the array of these labeled Morse code coefficients see A221915. See the Graham et al. reference, p. 302, on Euler's continuants and Morse code.
Row sums Q(n,1) = A001040(n+1), n >= 0. Alternating row sums Q(n,-1) = A058797(n). (End)
For fixed x the limit of the continued fraction K_{k>=1}(x/k) (see above) can be computed from the large order n behavior of Phat(n,x) and Q(n,x) given in the formula section in terms of Bessel functions. This follows with the well-known large n behavior of BesselI and BesselK, as given, e.g., in the Sidi and Hoggan reference, eqs. (1.1) and (1.2). See also the book by Olver, ch. 10, 7, p. 374. This continued fraction converges for fixed x to sqrt(x)*BesselI(1,2*sqrt(x))/BesselI(0,2*sqrt(x)). - Wolfdieter Lang, Mar 07 2013

			The irregular triangle a(n,m) begins:
  n\m          0          1        2        3      4    5  6 ...
  O:           1
  1:           1
  2:           2          1
  3:           6          4
  4:          24         18        1
  5:         120         96        9
  6:         720        600       72        1
  7:        5040       4320      600       16
  8:       40320      35280     5400      200      1
  9:      362880     322560    52920     2400     25
  10:    3628800    3265920   564480    29400    450    1
  11:   39916800   36288000  6531840   376320   7350   36
  12:  479001600  439084800 81648000  5080320 117600  882  1
...Reformatted and extended by _Wolfdieter Lang_, Mar 02 2013
E.g., to get row 7, multiply each term of row 6 by 7, then add the term NW of term in row 6: 5040 = (7)(720); 4320 = (7)(600) + 20; 600 = (7)(72) + 96; 16 = (7)(1) + 9. Thus row 7 = 5040 4320 600 16 with a sum of 9976 = a(7) of A001040.
From _Wolfdieter Lang_, Mar 02 2013: (Start)
Recurrence (short version): a(7,2) = 7*72 + 96 = 600.
Recurrence (long version): a(7,2) = (2*5-1)*72 + 96 - (5-1)^2*9 = 600.
a(7,2) = binomial(5,2)*5!/2! = 10*3*4*5 = 600. (End)

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, 2nd ed.; Addison-Wesley, 1994.
F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974 (1991 5th printing).

G. C. Greubel, Rows, n=0..149 of triangle, flattened
Avram Sidi and Philip E. Hogan, Asymptotics of modified Bessel functions of high order. Int. J. of Pure and Appl. Maths. 71 No. 3 (2011) 481-498.

Cf. A001040, A180047, A180048, A180049.
Cf. A021009 (Laguerre triangle). For the A-numbers of the column sequences see the Cf. section of A021009. A221913.
Cf. A052119.

L := (n, k) -> abs(coeff(n!*simplify(LaguerreL(n,x)), x, k)):
seq(seq(L(n-k, k), k=0..n/2), n=0..12); # Peter Luschny, Jan 22 2020

Table[CoefficientList[Denominator[Together[Fold[w/(#2+#1) &, Infinity, Reverse @ Table[k,{k,1,n}]]]],w],{n,16}]; (* Wouter Meeussen, Aug 08 2010 *)
(* or equivalently: *)
Table[( (n-m)!*Binomial[n-m, m] )/m! ,{n,0,15}, {m,0,Floor[n/2]}] (* Wouter Meeussen, Aug 08 2010 *)
row[n_] := If[n == 0, 1, x/ContinuedFractionK[x, i, {i, 0, n}] // Simplify // Together // Denominator // CoefficientList[#, x] &];
row /@ Range[0, 12] // Flatten (* Jean-François Alcover, Oct 28 2019 *)

a(n, m) = ((n-m)!/m!)*binomial(n-m,m). - Wouter Meeussen, Aug 08 2010
From Wolfdieter Lang, Mar 02 2013: (Start)
Recurrence (short version): a(n,m) = n*a(n-1,m) + a(n-2,m-1), n>=1, a(0,0) =1, a(n,-1) = 0, a(n,m) = 0 if n < 2*m. From the recurrence for the Q(n,x) polynomials given in a comment above.
Recurrence (long version): a(n,m) = (2*(n-m)-1)*a(n-1,m) + a(n-2,m-1) - (n-m-1)^2*a(n-2,m), n >= 1, a(0,0) =1, a(n,-1) = 0,  a(n,m) =0 if n < 2*m. From the standard three term recurrence for the unsigned orthogonal Laguerre polynomials. This recurrence can be simplified to the preceding one, because of the explicit factorial formula given above which follows from the one for the Laguerre coefficients (which, in turn, derives from the Rodrigues formula and the Leibniz rule). This proves the relation a(n,m) = |Lhat(n-m,m)|, with the coefficients |Lhat(n,m)| = |A021009(n,m)| of the unsigned n!*L(n,x) Laguerre polynomials.
For the e.g.f.s of the column sequences see  A021009 (here with different offset, which could be obtained by integration).
E.g.f. for row polynomials gQ(z,x) := Sum_{z>=0} Q(n,x)*z^n = (i*Pi*sqrt(x)/sqrt(1-z))*(BesselJ(1, 2*i*sqrt(x)*sqrt(1-z))*BesselY(0, 2*i*sqrt(x)) - BesselY(1, 2*i*sqrt(x)*sqrt(1-z))*BesselJ(0,2*i*sqrt(x))), with the imaginary unit i = sqrt(-1) and Bessel functions. (End)
The row polynomials are Q(n,x) = Pi*(z/2)^(n+1)*(BesselY(0,z)*BesselJ(n+1,z) - BesselJ(0,z)*BesselY(n+1,z)) with z := -i*2*sqrt(x), and the imaginary unit i. An alternative form is Q(n,x) = 2*(w/2)^(n+1)*(BesselI(0,w)*BesselK(n+1,w) - BesselK(0,w)*BesselI(n+1,w)*(-1)^(n+1)) with w := -2*sqrt(x). See A221913 for the derivation based on Abramowitz-Stegun's Handbook. - Wolfdieter Lang, Mar 06 2013
Lim_{n -> infinity} Q(n,x)/n! = BesselI(0,2*sqrt(x)). See a comment on asymptotics above. - Wolfdieter Lang, Mar 07 2013

Rows 12 to 17 added based on formula by Wouter Meeussen, Aug 08 2010

Name changed by Wolfdieter Lang, Mar 02 2013

A062140 Coefficient triangle of generalized Laguerre polynomials n!*L(n,4,x) (rising powers of x).

Original entry on oeis.org

1, 5, -1, 30, -12, 1, 210, -126, 21, -1, 1680, -1344, 336, -32, 1, 15120, -15120, 5040, -720, 45, -1, 151200, -181440, 75600, -14400, 1350, -60, 1, 1663200, -2328480, 1164240, -277200, 34650, -2310, 77, -1, 19958400, -31933440

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,4,x)= sum(a(n,m)*x^m,m=0..n) have g.f. exp(-z*x/(1-z))/(1-z)^5. They are Sheffer polynomials satisfying the binomial convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), with polynomials p(n,x)=sum(|A008297(n,m)|*(-x)^m, m=1..n) and p(0,x)=1 (for Sheffer polynomials see A048854 for S. Roman reference).

			Triangle begins:
  {1};
  {5,-1};
  {30,-12,1};
  {210,-126,21,-1};
  ...
2!*L(2,4,x)=30-12*x+x^2.

Indranil Ghosh, Rows 0..125
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001720(n+4), A062199, A062260-A062263. The row sums (signed) give A062265, the row sums (unsigned) give A062266.

Cf. A021009, A062137-A062139, A066667.

Flatten[Table[((-1)^m)*n!*Binomial[n+4,n-m]/m!,{n,0,11},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

row(n) = Vecrev(n!*pollaguerre(n, 4)); \\ Michel Marcus, Feb 06 2021

import math
f=math.factorial
def C(n,r):
    return f(n)//f(r)//f(n-r)
i=0
for n in range(26):
    for m in range(n+1):
        print(i, (-1)**m*f(n)*C(n+4,n-m)//f(m))
        i+=1 # Indranil Ghosh, Feb 23 2017

T(n, m) = ((-1)^m)*n!*binomial(n+4, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^5), m >= 0.

Previous Showing 11-20 of 57 results. Next

cp's OEIS Frontend

A105278 Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Haskell

Magma

Maple

Mathematica

Perl

Formula

Extensions

A132440 Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.

Views

Author

Keywords

Comments

Examples

References

Links

Programs

Maple

Mathematica

Formula

Extensions

A260832 a(n) = numerator(Jtilde2(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A009940 a(n) = n!*L_{n}(1), where L_{n}(x) is the n-th Laguerre polynomial.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

Extensions

A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A001809 a(n) = n! * n(n-1)/4.

Views

Author

Keywords

Comments

Examples

References