A048854 -id:A048854 - cp's OEIS frontend

A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

1, 2, 12, 120, 1680, 30240, 665280, 17297280, 518918400, 17643225600, 670442572800, 28158588057600, 1295295050649600, 64764752532480000, 3497296636753920000, 202843204931727360000, 12576278705767096320000, 830034394580628357120000, 58102407620643984998400000

Offset: 0

N. J. A. Sloane

Counts binary rooted trees (with out-degree <= 2), embedded in plane, with n labeled end nodes of degree 1. Unlabeled version gives Catalan numbers A000108.
Define a "downgrade" to be the permutation which places the items of a permutation in descending order. We are concerned with permutations that are identical to their downgrades. Only permutations of order 4n and 4n+1 can have this property; the number of permutations of length 4n having this property are equinumerous with those of length 4n+1. If a permutation p has this property then the reversal of this permutation also has it. a(n) = number of permutations of length 4n and 4n+1 that are identical to their downgrades. - Eugene McDonnell (eemcd(AT)mac.com), Oct 26 2003
Number of broadcast schemes in the complete graph on n+1 vertices, K_{n+1}. - Calin D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
Hankel transform is A137565. - Paul Barry, Nov 25 2009
The e.g.f. of 1/a(n) = n!/(2*n)! is (exp(sqrt(x)) + exp(-sqrt(x)) )/2. - Wolfdieter Lang, Jan 09 2012
From Tom Copeland, Nov 15 2014: (Start)
Aerated with intervening zeros (1,0,2,0,12,0,120,...) = a(n) (cf. A123023 and A001147), the e.g.f. is e^(t^2), so this is the base for the Appell sequence with e.g.f. e^(t^2) e^(x*t) = exp(P(.,x),t) (reverse A059344, cf. A099174, A066325 also). P(n,x) = (a. + x)^n with (a.)^n = a_n and comprise the umbral compositional inverses for e^(-t^2)e^(x*t) = exp(UP(.,x),t), i.e., UP(n,P(.,t)) = x^n = P(n,UP(.,t)), e.g., (P(.,t))^n = P(n,t).
Equals A000407*2 with leading 1 added. (End)
a(n) is also the number of square roots of any permutation in S_{4*n} whose disjoint cycle decomposition consists of 2*n transpositions. - Luis Manuel Rivera Martínez, Mar 04 2015
Self-convolution gives A076729. - Vladimir Reshetnikov, Oct 11 2016
For n > 1, it follows from the formula dated Aug 07 2013 that a(n) is a Zumkeller number (A083207). - Ivan N. Ianakiev, Feb 28 2017
For n divisible by 4, a(n/4) is the number of ways to place n points on an n X n grid with pairwise distinct abscissae, pairwise distinct ordinates, and 90-degree rotational symmetry. For n == 1 (mod 4), the number of ways is a((n-1)/4) because the center point can be considered "fixed". For 180-degree rotational symmetry see A006882, for mirror symmetry see A000085, A135401, and A297708. - Manfred Scheucher, Dec 29 2017

			The following permutations of order 8 and their reversals have this property:
  1 7 3 5 2 4 0 6
  1 7 4 2 5 3 0 6
  2 3 7 6 1 0 4 5
  2 4 7 1 6 0 3 5
  3 2 6 7 0 1 5 4
  3 5 1 7 0 6 2 4

D. E. Knuth, The Art of Computer Programming, Vol. 4, Section 7.2.1.6, Eq. 32.
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181.
Eugene McDonnell, "Magic Squares and Permutations" APL Quote-Quad 7.3 (Fall, 1976)
R. W. Robinson, Counting arrangements of bishops, pp. 198-214 of Combinatorial Mathematics IV (Adelaide 1975), Lect. Notes Math., 560 (1976).
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).

N. J. A. Sloane, Table of n, a(n) for n = 0..100
Marcello Artioli, Giuseppe Dattoli, Silvia Licciardi, and Rosa Maria Pidatella, Hermite and Laguerre Functions: a Unifying Point of View, Università degli Studi di Catania, Sicily, Italy (2019).
Murray Bremner and Martin Markl, Distributive laws between the Three Graces, arXiv:1809.08191 [math.AT], 2018.
R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015) # 15.3.2.
Peter J. Cameron, Some treelike objects, Quart. J. Math. Oxford Ser. (2) 38 (1987), no. 150, 155--183. MR0891613 (89a:05009). See p. 155. - _N. J. A. Sloane_, Apr 18 2014
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Elliot J. Carr and Matthew J. Simpson, New homogenization approaches for stochastic transport through heterogeneous media, arXiv:1810.08890 [physics.bio-ph], 2018.
W. Y. C. Chen, L. W. Shapiro and L. L. M. Young, Parity reversing involutions on plane trees and 2-Motzkin paths, arXiv:math/0503300 [math.CO], 2005.
Ali Chouria, Vlad-Florin Drǎgoi, and Jean-Gabriel Luque, On recursively defined combinatorial classes and labelled trees, arXiv:2004.04203 [math.CO], 2020.
P. Cvitanovic, Group theory for Feynman diagrams in non-Abelian gauge theories, Phys. Rev. D14 (1976), 1536-1553.
Nick Early, Honeycomb tessellations and canonical bases for permutohedral blades, arXiv:1810.03246 [math.CO], 2018.
John Engbers, David Galvin, and Clifford Smyth, Restricted Stirling and Lah numbers and their inverses, arXiv:1610.05803 [math.CO], 2016. See p. 8.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 127
S. Goodenough and C. Lavault, On subsets of Riordan subgroups and Heisenberg--Weyl algebra, arXiv preprint arXiv:1404.1894 [cs.DM], 2014.
S. Goodenough and C. Lavault, Overview on Heisenberg—Weyl Algebra and Subsets of Riordan Subgroups, The Electronic Journal of Combinatorics, 22(4) (2015), #P4.16,
H. W. Gould, A class of binomial sums and a series transform, Utilitas Math., 45 (1994), 71-83. (Annotated scanned copy)
A. M. Ibrahim, Extension of factorial concept to negative numbers, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 115
L. C. Larson, The number of essentially different nonattacking rook arrangements, J. Recreat. Math., 7 (No. 3, 1974), circa pages 180-181. [Annotated scan of pages 180 and 181 only]
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, Australas. J. Combin. 52 (2012), 41-54 (Theorem 1).
Jesús Leaños, Rutilo Moreno, and Luis Manuel Rivera-Martínez, On the number of mth roots of permutations, arXiv:1005.1531 [math.CO], 2010-2011.
Édouard Lucas, Théorie des Nombres, Gauthier-Villars, Paris, 1891, Vol. 1, p. 221.
R. J. Marsh and P. P. Martin, Pascal arrays: counting Catalan sets, arXiv:math/0612572 [math.CO], 2006.
Calin D. Morosan, On the number of broadcast schemes in networks, Information Processing Letters, Volume 100, Issue 5 (2006), 188-193.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.
J. Riordan, Letter to N. J. A. Sloane, Feb 03 1975 (with notes by njas)
H. E. Salzer, Coefficients for expressing the first thirty powers in terms of the Hermite polynomials, Math. Comp., 3 (1948), 167-169.
Elena L. Wang and Guoce Xin, On Ward Numbers and Increasing Schröder Trees, arXiv:2507.15654 [math.CO], 2025. See pp. 12-13.
Index to divisibility sequences
Index entries for related partition-counting sequences

Cf. A037224, A048854, A001147, A007696, A008545, A122670 (essentially the same sequence), A000165, A047055, A047657, A084947, A084948, A084949, A010050, A000142, A008275, A000108, A000984, A008276, A000680, A094216.
Catalan(n-1)*M^(n-1)*n! for M=1,2,3,4,5,6: A001813, A052714 (or A144828), A221954, A052734, A221953, A221955.
Cf. A123023, A059344, A099174, A066325, A001700, A000407, A006882.
Cf. A000085, A135401, A297708. - Manfred Scheucher, Jan 07 2018

List([0..20],n->Factorial(2*n)/Factorial(n)); # Muniru A Asiru, Nov 01 2018

[Factorial(2*n)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 09 2018

A001813 := n->(2*n)!/n!;
A001813 := n -> mul(k, k = select(k-> k mod 4 = 2,[$1 .. 4*n])):
seq(A001813(n), n=0..16);  # Peter Luschny, Jun 23 2011

Table[(2n)!/n!, {n,0,20}] (* Harvey P. Dale, May 02 2011 *)

makelist(binomial(n+n, n)*n!,n,0,30); /* Martin Ettl, Nov 05 2012 */

a(n)=binomial(n+n,n)*n! \\ Charles R Greathouse IV, Jun 15 2011

first(n) = x='x+O('x^n); Vec(serlaplace((1 - 4*x)^(-1/2))) \\ Iain Fox, Jan 01 2018 (corrected by Iain Fox, Jan 11 2018)

from math import factorial
def A001813(n): return factorial(n<<1)//factorial(n) # Chai Wah Wu, Feb 14 2023

[binomial(2*n,n)*factorial(n) for n in range(0, 17)] # Zerinvary Lajos, Dec 03 2009

E.g.f.: (1-4*x)^(-1/2).
a(n) = (2*n)!/n! = Product_{k=0..n-1} (4*k + 2) = A081125(2*n).
Integral representation as n-th moment of a positive function on a positive half-axis: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/(sqrt(x)*2*sqrt(Pi)) dx, n >= 0. This representation is unique. - Karol A. Penson, Sep 18 2001
Define a'(1)=1, a'(n) = Sum_{k=1..n-1} a'(n-k)*a'(k)*C(n, k); then a(n)=a'(n+1). - Benoit Cloitre, Apr 27 2003
With interpolated zeros (1, 0, 2, 0, 12, ...) this has e.g.f. exp(x^2). - Paul Barry, May 09 2003
a(n) = A000680(n)/A000142(n)*A000079(n) = Product_{i=0..n-1} (4*i + 2) = 4^n*Pochhammer(1/2, n) = 4^n*GAMMA(n+1/2)/sqrt(Pi). - Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003
For asymptotics, see the Robinson paper.
a(k) = (2*k)!/k! = Sum_{i=1..k+1} |A008275(i,k+1)| * k^(i-1). - André F. Labossière, Jun 21 2007
a(n) = 12*A051618(a) n >= 2. - Zerinvary Lajos, Feb 15 2008
a(n) = A000984(n)*A000142(n). - Zerinvary Lajos, Mar 25 2008
a(n) = A016825(n-1)*a(n-1). - Roger L. Bagula, Sep 17 2008
a(n) = (-1)^n*A097388(n). - D. Morosan (cd_moros(AT)alumni.concordia.ca), Nov 28 2008
From Paul Barry, Jan 15 2009: (Start)
G.f.: 1/(1-2x/(1-4x/(1-6x/(1-8x/(1-10x/(1-... (continued fraction);
a(n) = (n+1)!*A000108(n). (End)
a(n) = Sum_{k=0..n} A132393(n,k)*2^(2n-k). - Philippe Deléham, Feb 10 2009
G.f.: 1/(1-2x-8x^2/(1-10x-48x^2/(1-18x-120x^2/(1-26x-224x^2/(1-34x-360x^2/(1-42x-528x^2/(1-... (continued fraction). - Paul Barry, Nov 25 2009
a(n) = A173333(2*n,n) for n>0; cf. A006963, A001761. - Reinhard Zumkeller, Feb 19 2010
From Gary W. Adamson, Jul 19 2011: (Start)
a(n) = upper left term of M^n, M = an infinite square production matrix as follows:
  2, 2, 0, 0, 0, 0, ...
  4, 4, 4, 0, 0, 0, ...
  6, 6, 6, 6, 0, 0, ...
  8, 8, 8, 8, 8, 0, ...
  ...
(End)
a(n) = (-2)^n*Sum_{k=0..n} 2^k*s(n+1,n+1-k), where s(n,k) are the Stirling numbers of the first kind, A048994. - Mircea Merca, May 03 2012
G.f.: 1/Q(0), where Q(k) = 1 + x*(4*k+2) - x*(4*k+4)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 18 2013
G.f.: 2/G(0), where G(k) = 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 8*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 2*x/(2*x + 1/(2*k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 01 2013
D-finite with recurrence: a(n) = (4*n-6)*a(n-2) + (4*n-3)*a(n-1), n>=2. - Ivan N. Ianakiev, Aug 07 2013
Sum_{n>=0} 1/a(n) = (exp(1/4)*sqrt(Pi)*erf(1/2) + 2)/2 = 1 + A214869, where erf(x) is the error function. - Ilya Gutkovskiy, Nov 10 2016
Sum_{n>=0} (-1)^n/a(n) = 1 - sqrt(Pi)*erfi(1/2)/(2*exp(1/4)), where erfi(x) is the imaginary error function. - Amiram Eldar, Feb 20 2021
a(n) = 1/([x^n] hypergeom([1], [1/2], x/4)). - Peter Luschny, Sep 13 2024
a(n) = 2^n*n!*JacobiP(n, -1/2, -n, 3). - Peter Luschny, Jan 22 2025
G.f.: 2F0(1,1/2;;4x). - R. J. Mathar, Jun 07 2025

More terms from James Sellers, May 01 2000

A091042 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.

Original entry on oeis.org

1, 1, 3, 1, 10, 5, 1, 21, 35, 7, 1, 36, 126, 84, 9, 1, 55, 330, 462, 165, 11, 1, 78, 715, 1716, 1287, 286, 13, 1, 105, 1365, 5005, 6435, 3003, 455, 15, 1, 136, 2380, 12376, 24310, 19448, 6188, 680, 17, 1, 171, 3876, 27132, 75582, 92378, 50388, 11628, 969, 19, 1, 210, 5985, 54264, 203490, 352716, 293930, 116280, 20349, 1330, 21

Offset: 0

Wolfdieter Lang, Jan 23 2004

The row polynomials Pe(n, x) := Sum_{m=0..n} a(n, m)*x^m appear as numerators of the generating functions for the even numbered column sequences of array A034870.
Elements have the same parity as those of Pascal's triangle.
All zeros of polynomial Pe(n, x) are negative. They are -tan^2(Pi/2*n+1), -tan^2(2*Pi/2*n+1), ..., -tan^2(n*Pi/2*n+1). Moreover, for m >= 1, Pe(m, -x^2) is the characteristic polynomial of the linear difference equation with constant coefficients for differences between multiples of 2*m+1 with even and odd digit sum in base 2*m in the interval [0,(2*m)^n). - Vladimir Shevelev and Peter J. C. Moses, May 22 2012
Row reverse of A103327. - Peter Bala, Jul 29 2013
The row polynomial Pe(d, x), multiplied by (2*d)!/d! = A001813(d), gives the numerator polynomial of the o.g.f. of the sequence of the diagonal d, for d >= 0, of the Sheffer triangle Lah[4,1] given in A048854. - Wolfdieter Lang, Oct 12 2017

			Triangle a(n, m) begins:
n\m  0   1    2     3      4      5      6      7     8    9  10 ...
0:   1
1:   1   3
2:   1  10    5
3:   1  21   35     7
4:   1  36  126    84      9
5:   1  55  330   462    165     11
6:   1  78  715  1716   1287    286     13
7:   1 105 1365  5005   6435   3003    455     15
8:   1 136 2380 12376  24310  19448   6188    680    17
9:   1 171 3876 27132  75582  92378  50388  11628   969   19
10:  1 210 5985 54264 203490 352716 293930 116280 20349 1330  21
... reformatted and extended. - _Wolfdieter Lang_, Oct 12 2017
From _Peter Bala_, Jan 30 2022: (Start)
(1/2)*(N^2 + N) = Sum_{j = 1..N} j.
(1/2)*(N^2 + N)^3 = Sum_{j = 1..N} j^3 + 3*Sum_{j = 1..N} j^5.
(1/2)*(N^2 + N)^5 = Sum_{j = 1..N} j^5 + 10*Sum_{j = 1..N} j^7 + 5*Sum_{j = 1..N} j^9.
(1/2)*(N^2 + N)^7 = Sum_{j = 1..N} j^7 + 21*Sum_{j = 1..N} j^9 + 35*Sum_{j = 1..N} j^11 + 7*Sum_{j = 1..N} j^13. (End)

A. M. Yaglom and I. M. Yaglom, An elementary proof of the Wallis, Leibniz and Euler formulas for pi. Uspekhi Matem. Nauk, VIII (1953), 181-187(in Russian).

Indranil Ghosh, Rows 0..120 of triangle, flattened
Wolfdieter Lang, First 9 rows.
V. Shevelev, On monotonic strengthening of Newman-like phenomenon on (2m+1)-multiples in base 2m, arXiv:0710.3177 [math.NT], 2007.
V. Shevelev and P. Moses, Tangent power sums and their applications, arXiv:1207.0404 [math.NT], 2012-2014.

Cf. A212500, A038754. A000302 (row sums), A085478, A103327 (row reverse), A048854, A103328.

Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n+1, 2*k) ))); # G. C. Greubel, Aug 01 2019

[[Binomial(2*n+1,2*k): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Aug 01 2019

f := (x, t) -> cosh(sqrt(x)*t)*sinh(t); seq(seq(coeff(((2*n+1)!*coeff(series(f(x,t),t,2*n+2),t,2*n+1)),x,k),k=0..n),n=0..9); # Peter Luschny, Jul 29 2013

T[n_, k_] /; 0 <= k <= n := T[n, k] = 2T[n-1, k] + 2T[n-1, k-1] + 2T[n-2, k-1] - T[n-2, k] - T[n-2, k-2]; T[0, 0] = T[1, 0] = 1; T[1, 1] = 3; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* Jean-François Alcover, Jul 29 2018, after Philippe Deléham *)
Table[Binomial[2*n+1, 2*k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2019 *)

T(n,k) = binomial(2*n+1, 2*k); \\ G. C. Greubel, Aug 01 2019

from math import comb, isqrt
def A091042(n): return comb((r:=(m:=isqrt(k:=n+1<<1))-(k<=m*(m+1)))<<1|1,n-comb(r+1,2)<<1) # Chai Wah Wu, Apr 30 2025

[[binomial(2*n+1, 2*k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019

T(n, m) = binomial(2*n+1, 2*m) = A007318(2*n+1, 2*m), n >= m >= 0, otherwise 0.
From Peter Bala, Jul 29 2013: (Start)
E.g.f.: sinh(t)*cosh(sqrt(x)*t) = t + (1 + 3*x)*t^3/3! + (1 + 10*x + 5*x^2)*t^5/5! + (1 + 21*x + 35*x^2 + 7*x^3)*t^7/7! + ....
O.g.f.: A(x,t) = (1 + (x - 1)*t)/( (1 + (x - 1)*t)^2 - 4*t*x ) = 1 + (1 + 3*x)*t + (1 + 10*x + 5*x^2)*t^2 + ...
The function A( x/(x + 4), t*(x + 4)/4 ) = 1 + (1 + x)*t + (1 + 3*x + x^2)*t^2 + ... is the o.g.f. for A085478.
O.g.f. for n-th diagonal: ( Sum_{k = 0..n} binomial(2*n,2*k)*x^k )/(1 - x)^(2*n).
n-th row polynomial R(n,x) = (1/2)*( (1 + sqrt(x))^(2*n+1) - (sqrt(x) - 1)^(2*n+1) ).
Row sums A000302. (End)
T(n, k) = 2*T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-1) - T(n-2,k) - T(n-2,k-2) with T(0,0)=T(1,0)=1, T(1,1)=3, T(n,k)=0 if k < 0 or if k > n. - Philippe Deléham, Nov 26 2013
From Peter Bala, Jan 31 2022: (Start)
Define S(r,N) = Sum_{j = 1..N} j^r. Then the following identity holds for n >= 0: (1/2)*(N^2 + N)^(2*n+1) = T(n,0)*S(2*n+1,N) + T(n,1)*S(2*n+3,N) + ... + T(n,n)* S(4*n+1,N). Some examples are given below. (End)

A196838 Numerators of coefficients of Bernoulli polynomials with rising powers of the variable.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, 0, 1, -3, 1, -1, 0, 1, -2, 1, 0, -1, 0, 5, -5, 1, 1, 0, -1, 0, 5, -3, 1, 0, 1, 0, -7, 0, 7, -7, 1, -1, 0, 2, 0, -7, 0, 14, -4, 1, 0, -3, 0, 2, 0, -21, 0, 6, -9, 1, 5, 0, -3, 0, 5, 0, -7, 0, 15, -5, 1, 0, 5, 0, -11, 0, 11, 0, -11, 0, 55, -11, 1

Offset: 0

Wolfdieter Lang, Oct 23 2011

The denominator triangle is found under A196839.
This is the row reversed triangle A053382.
From Wolfdieter Lang, Oct 25 2011: (Start)
This is the Sheffer triangle (z/(exp(z)-1),z), meaning that the column e.g.f.'s are as given below in the formula section. In Roman's book `The Umbral Calculus`, Ch. 2, 5., p. 26ff this is called Appell for (exp(t)-1)/t (see A048854 for the reference).
The e.g.f. for the a- and z-sequence for this Sheffer triangle is 1 and (x-exp(x)+1)/x^2, respectively. See the link under A006232 for the definition. The z-sequence is z(n) = -1/(2*A000217(n+1)). This leads to the recurrence relations given below.
The e.g.f. for the row sums is x/(1-exp(-x)), leading to the rational sequence A164555(n)/A027664(n). The e.g.f. of the alternating row sums is
  x/(exp(x)*(exp(x)-1)), leading to the rational sequence
  (-1)^n*A164558(n)/A027664(n).
(End)

			The triangle starts with
n\m 0  1  2  3  4  5  6  7  8 ...
0:  1
1: -1  1
2:  1 -1  1
3:  0  1 -3  1
4: -1  0  1 -2  1
5:  0 -1  0  5 -5  1
6:  1  0 -1  0  5 -3  1
7:  0  1  0 -7  0  7 -7  1
8: -1  0  2  0 -7  0 14 -4  1
...
The rational triangle a(n,m)/A196839(n,m) starts with:
n\m   0     1     2    3    4    5     6    7   8 ...
0:    1
1:  -1/2    1
2:   1/6   -1     1
3:    0    1/2  -3/2   1
4:  -1/30   0     1   -2    1
5:    0   -1/6    0   5/3 -5/2   1
6:   1/42   0   -1/2   0   5/2  -3     1
7:    0    1/6    0  -7/6   0   7/2  -7/2   1
8:  -1/30   0    2/3   0  -7/3   0   14/3  -4   1
...
E.g., Bernoulli(2,x) = (1/6)*x^0 - 1*x^1 + 1*x^2.

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1991 (Seventh printing).Second ed. 1994.

Naho Kawasaki and Yasuo Ohno, The triangle algorithm for Bernoulli polynomials, Integers, vol. 23 (2023). (See figure 4.)
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
D. H. Lehmer, A new approach to Bernoulli polynomials, The American mathematical monthly 95.10 (1988): 905-911.

Three versions of coefficients of Bernoulli polynomials: A053382/A053383; for reflected version see A196838/A196839; see also A048998 and A048999.

# Without using Maple's Bernoulli polynomials (Kawasaki and Ohno call it
# the 'triangle algorithm for B(n, x)'):
b := proc(n, m, x) option remember; if n = 0 then 1/(m + 1) else
normal((m + 1)*b(n-1, m + 1, x) - (m + 1 - x)*b(n-1, m, x)) fi end:
Bcoeffs := n -> local k; [seq(coeff(b(n, 0, x), x, k), k = 0..n)]:
for n from 0 to 8 do numer(Bcoeffs(n)) od; # Peter Luschny, Jun 16 2023

row[n_] := CoefficientList[BernoulliB[n, x], x] // Numerator;
Table[row[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 15 2018 *)

row(n) = apply(x->numerator(x), Vecrev(bernpol(n)));
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Jun 15 2018

T(n,m) = numerator([x^m]Bernoulli(n,x)), n>=0, m=0..n.
E.g.f. of Bernoulli(n,x): z*exp(x*z)/(exp(z)-1).
See the Graham et al. reference, eq. (7.80), p. 354.
From Wolfdieter Lang, Oct 25 2011: (Start)
The e.g.f. for column no. m>=0 of the rational triangle B(n,m):=a(n,m)/A096839(n,m) is x^(m+1)/(m!*(exp(x)-1)).
(see the Sheffer-Appell comment above).
The Sheffer a-sequence, given as comment above, leads to the recurrence r(n,m)=(n/m)*r(n-1,m-1), n>=1, m>=1. E.g., -1/6 = B(5,1) = (5/1)*B(4,0)= -5/30 = -1/6.
The Sheffer z-sequence, given as comment above, leads to the recurrence
B(n,0) = n*sum(z(j)*B(n-1,j),j=0..n-1), n>=1. B(0,0)=1.
E.g., -1/30 = B(4,0) = 4*((-1/2)*0 + (-1/6)*(1/2) + (-1/12)*(-3/2) + (-1/20)*1) = -1/30.
(End)
T(n,m) = numerator(binomial(n,m)*Bernoulli(n-m)). - Fabián Pereyra, Mar 04 2020

A049444 Generalized Stirling number triangle of first kind.

Original entry on oeis.org

1, -2, 1, 6, -5, 1, -24, 26, -9, 1, 120, -154, 71, -14, 1, -720, 1044, -580, 155, -20, 1, 5040, -8028, 5104, -1665, 295, -27, 1, -40320, 69264, -48860, 18424, -4025, 511, -35, 1, 362880, -663696, 509004, -214676, 54649, -8624, 826, -44, 1, -3628800, 6999840, -5753736

Offset: 0

Wolfdieter Lang

T(n, k) = ^2P_n^k in the notation of the given reference with T(0, 0) := 1. The monic row polynomials s(n,x) := Sum_{m=0..n} T(n, k)*x^k which are s(n, x) = Product_{j=0..n-1} (x-(2+j)), n >= 1 and s(0, x)=1 satisfy s(n, x+y) = Sum_{k=0..n} binomial(n, k)*s(k,x)*S1(n-k, y), with the Stirling1 polynomials S1(n, x) = Sum_{m=1..n} (A008275(n, m)*x^m) and S1(0, x)=1.
In the umbral calculus (see the S. Roman reference given in A048854) the s(n, x) polynomials are called Sheffer polynomials for (exp(2*t), exp(t)-1). This translates to the usual exponential Riordan (Sheffer) notation (1/(1+x)^2, log(1+x)).
See A143491 for the unsigned version of this array and A143494 for the inverse. - Peter Bala, Aug 25 2008
Corresponding to the generalized Stirling number triangle of second kind A137650. - Peter Luschny, Sep 18 2011
Unsigned, reversed rows (cf. A145324, A136124) are the dimensions of the cohomology of a complex manifold with a symmetric group (S_n) action. See p. 17 of the Hyde and Lagarias link. See also the Murri link for an interpretation as the Betti numbers of the moduli space M(0,n) of smooth Riemann surfaces. - Tom Copeland, Dec 09 2016
The row polynomials s(n, x) = (-1)^n*risingfactorial(2 - x, n) are related to the column sequences of the unsigned Abel triangle A137452(n, k), for k >= 2. See the formula there. - Wolfdieter Lang, Nov 21 2022

			The Triangle  begins:
n\k       0       1        2       3       4      5      6    7   8 9 ...
0:        1
1:       -2       1
2:        6      -5        1
3:      -24      26       -9       1
4:      120    -154       71     -14       1
5      -720    1044     -580     155     -20      1
6:     5040   -8028     5104   -1665     295    -27      1
7:   -40320   69264   -48860   18424   -4025    511    -35    1
8:   362880 -663696   509004 -214676   54649  -8624    826  -44
9: -3628800 6999840 -5753736 2655764 -761166 140889 -16884 1266 -54 1
...  [reformatted by _Wolfdieter Lang_, Nov 21 2022]

Y. Manin, Frobenius Manifolds, Quantum Cohomology and Moduli Spaces, American Math. Soc. Colloquium Publications Vol. 47, 1999. [From Tom Copeland, Jun 29 2008]
S. Roman, The Umbral Calculus, Academic Press, 1984 (also Dover Publications, 2005).

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
E. Getzler, Operads and moduli spaces of genus 0 Riemann surfaces, arXiv:alg-geom/9411004, 1994, (see p. 23, g(x,t)). [From _Tom Copeland_, Dec 11 2011]
T. Hyde and J. Lagarias Polynomial splitting measures and cohomology of the pure braid group, arXiv preprint arXiv:1604.05359 [math.RT], 2016.
Y. Manin, Generating functions in algebraic geometry and sums over trees, arXiv:alg-geom/9407005, 1994, (Eqn. 0.7 and 1.7). [From _Tom Copeland_, Dec 10 2011]
D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
R. Murri, Fatgraph Algorithms and the Homology of the Kontsevich Complex, arXiv:1202.1820 [math.AG], 2012, (see Table 1, p. 3). [From _Tom Copeland_, Sep 18 2012]

Unsigned column sequences are A000142(n+1), A001705-A001709. Row sums (signed triangle): n!*(-1)^n, row sums (unsigned triangle): A001710(n-2). Cf. A008275 (Stirling1 triangle).
Cf. A000035, A084938, A094645, A094646.
Cf. A143491, A143494.
Cf. A136124, A137452, A137650.

a049444 n k = a049444_tabl !! n !! k
a049444_row n = a049444_tabl !! n
a049444_tabl = map fst $ iterate (\(row, i) ->
   (zipWith (-) ([0] ++ row) $ map (* i) (row ++ [0]), i + 1)) ([1], 2)
-- Reinhard Zumkeller, Mar 11 2014

A049444_row := proc(n) local k,i;
add(add(Stirling1(n, n-i), i=0..k)*x^(n-k-1),k=0..n-1);
seq(coeff(%,x,k),k=1..n-1) end:
seq(print(A049444_row(n)),n=1..7); # Peter Luschny, Sep 18 2011
A049444:= (n, k)-> add((-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k), j=0..n):
seq(print(seq(A049444(n, k), k=0..n)), n=0..11);  # Mélika Tebni, May 02 2022

t[n_, i_] = Sum[(-1)^k*Binomial[n, k]*(k+1)!*StirlingS1[n-k, i], {k, 0, n-i}]; Flatten[Table[t[n, i], {n, 0, 9}, {i, 0, n}]] [[1 ;; 48]]
(* Jean-François Alcover, Apr 29 2011, after Milan Janjic *)

T(n, k) = T(n-1, k-1) - (n+1)*T(n-1, k), n >= k >= 0; T(n, k) = 0, n < k; T(n, -1) = 0, T(0, 0) = 1.
E.g.f. for k-th column of signed triangle: ((log(1+x))^k)/(k!*(1+x)^2).
Triangle (signed) = [-2, -1, -3, -2, -4, -3, -5, -4, -6, -5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...]; triangle (unsigned) = [2, 1, 3, 2, 4, 3, 5, 4, 6, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...], where DELTA is Deléham's operator defined in A084938 (unsigned version in A143491).
E.g.f.: (1 + x)^(y-2). - Vladeta Jovovic, May 17 2004 [For row polynomials s(n, y)]
With P(n, t) = Sum_{j=0..n-2} T(n-2,j) * t^j and P(1, t) = -1 and P(0, t) = 1, then G(x, t) = -1 + exp[P(.,t)*x] = [(1+x)^t - 1 - t^2 * x] / [t(t-1)], whose compositional inverse in x about 0 is given in A074060. G(x, 0) = -log(1+x) and G(x, 1) = (1+x) log(1+x) - 2x. G(x, q^2) occurs in formulas on pages 194-196 of the Manin reference. - Tom Copeland, Feb 17 2008
If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j), then T(n,i) = f(n,i,2), for n=1,2,...; i=0..n. - Milan Janjic, Dec 21 2008
T(n, k) = Sum_{j=0..n} (-1)^(n-j)*(n-j+1)!*binomial(n, j)*Stirling1(j, k). - Mélika Tebni, May 02 2022
From Wolfdieter Lang, Nov 24 2022: (Start)
Recurrence for row polynomials {s(n, x)}_{n>=0}: s(0, x) = 1, s(n, x) = (x - 2)*exp(-(d/dx)) s(n-1, x), for n >= 1. This is adapted from the general Sheffer result given by S. Roman, Corollary 3.7.2., p. 50.
Recurrence for column sequence {T(n, k)}{n>=k}: T(n, n) = 1, T(n, k) = (n!/(n-k))*Sum{j=k..n-1} (1/j!)*(a(n-1-j) + k*beta(n-1-j))*T(n-1, k), for k >= 0, where alpha = repeat(-2, 2) and beta(n) = [x^n] (d/dx)log(log(x)/x) = (-1)^(n+1)*A002208(n+1)/A002209(n+1), for n >= 0. This is the adapted Boas-Buck recurrence, also given in Rainville, Theorem 50., p. 141, For the references and a comment see A046521. (End)

Second formula corrected by Philippe Deléham, Nov 09 2008

A062137 Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).

Original entry on oeis.org

1, 4, -1, 20, -10, 1, 120, -90, 18, -1, 840, -840, 252, -28, 1, 6720, -8400, 3360, -560, 40, -1, 60480, -90720, 45360, -10080, 1080, -54, 1, 604800, -1058400, 635040, -176400, 25200, -1890, 70, -1, 6652800, -13305600, 9313920

Offset: 0

Wolfdieter Lang, Jun 19 2001

The row polynomials s(n,x) := n!*L(n,3,x) = Sum_{m=0..n} a(n,m)*x^m have e.g.f. exp(-z*x/(1-z))/(1-z)^4. 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).
These polynomials appear in the radial part of the l=1 (p-wave) eigen functions for the discrete energy levels of the H-atom. See Messiah reference.
The unsigned version of this triangle is the triangle of unsigned 2-Lah numbers A143497. - Peter Bala, Aug 25 2008
This matrix (unsigned) is embedded in that for n!*L(n,-3,-x). Introduce 0,0,0 to each unsigned row and then add 1,-2,1,4,2,1 to the beginning of the array as the first three rows to generate n!*L(n,-3,-x). - Tom Copeland, Apr 21 2014
From Wolfdieter Lang, Jul 07 2014: (Start)
The standard Rodrigues formula for the monic generalized Laguerre polynomials (also called Laguerre-Sonin polynomials) is Lm(n,alpha,x) := (-1)^n*n!*L(n,3,x) is x^(-alpha)*exp(x)*(d/dx)^n(exp(-x)*x^(n+alpha)).
Another Rodrigues type formula is Lm(n,alpha,x) = exp(x)*x^(-alpha+n+1)*(-x^2*d/dx)^n*(exp(-x)*x^(alpha+1)). This is derivable from the differential - difference relation of Lm(n,alpha,x) which yields first the creation operator formula -(x*d/dx + (-x + alpha + n + 1))*Lm(n,alpha,x) = Lm(n+1,alpha,x) or in the variable q = log(x) the operator -(d/dq + alpha + n + 1 - exp(q)).
The identity (due to Christoph Mayer) (d/dq - (d/dq)W(q))*f(q) = exp(W(q)*d/dq(exp(-W(q)*f(q)) is then iterated with W(q) = W(alpha,n,q) = exp(q) - (alpha + n + 1)*q and a further change of variables leads then to the given result. (End)

			The triangle a(n,m) begins:
n\m       0        1       2     3    4   5 ...
0:        1
1:        4       -1
2:       20      -10      1
3:      120      -90     18     -1
4:      840     -840    252    -28    1
5:     6720    -8400   3360   -560   40  -1
... Formatted by _Wolfdieter Lang_, Jul 07 2014
For more rows see the link.
n = 2: 2!*L(2,3,x) = 20 - 10*x + x^2.

A. Messiah, Quantum mechanics, vol. 1, p. 419, eq.(XI.18a), North Holland, 1969.

Indranil Ghosh, Rows 0..125, flattened
Wolfdieter Lang, First eleven rows of the triangle.
Index entries for sequences related to Laguerre polynomials

For m=0..5 the (unsigned) columns give A001715, A061206, A062141-A062144. The row sums (signed) give A062146, the row sums (unsigned) give A062147.
Cf. A143497. - Peter Bala, Aug 25 2008
Cf. A062139, A105278. - Wolfdieter Lang, Jul 07 2014

Flatten[Table[((-1)^m)*n!*Binomial[n+3,n-m]/m!,{n,0,9},{m,0,n}]] (* Indranil Ghosh, Feb 23 2017 *)

row(n) = Vecrev(n!*pollaguerre(n, 3)); \\ Michel Marcus, Feb 06 2021

a(n, m) = ((-1)^m)*n!*binomial(n+3, n-m)/m!.

E.g.f. for m-th column sequence: ((-x/(1-x))^m)/(m!*(1-x)^4), m >= 0.

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)

A111595 Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).

Original entry on oeis.org

1, 0, 1, 1, -2, 1, 0, 9, -6, 1, 9, -36, 42, -12, 1, 0, 225, -300, 130, -20, 1, 225, -1350, 2475, -1380, 315, -30, 1, 0, 11025, -22050, 15435, -4620, 651, -42, 1, 11025, -88200, 220500, -182280, 67830, -12600, 1204, -56, 1, 0, 893025, -2381400, 2302020, -1020600, 235494, -29736, 2052, -72

Offset: 0

Wolfdieter Lang, Aug 23 2005

This is a Sheffer triangle (lower triangular exponential convolution matrix). For Sheffer row polynomials see the S. Roman reference and explanations under A048854.
In the umbral notation of the S. Roman reference this would be called Sheffer for ((sqrt(1-2*t))/(1-t), t/(1-t)).
The associated Sheffer triangle is A111596.
Matrix logarithm equals A112239. - Paul D. Hanna, Aug 29 2005
The row polynomials (1/2^n)* H(n,sqrt(x/2))^2, with the Hermite polynomials H(n,x), have e.g.f. (1/sqrt(1-y^2))*exp(x*y/(1+y)).
The row polynomials s(n,x):=sum(a(n,m)*x^m,m=0..n), together with the associated row polynomials p(n,x) of A111596, satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
The unsigned column sequences are: A111601, A111602, A111777-A111784, for m=1..10.

			The triangle a(n, m) begins:
n\m       0         1         2          3         4         5       6       7     8    9  10 ...
0:        1
1:        0         1
2:        1        -2         1
3:        0         9        -6          1
4:        9       -36        42        -12         1
5:        0       225      -300        130       -20         1
6:      225     -1350      2475      -1380       315       -30       1
7:        0     11025    -22050      15435     -4620       651     -42       1
8:    11025    -88200    220500    -182280     67830    -12600    1204     -56     1
9:        0    893025  -2381400    2302020  -1020600    235494  -29736    2052   -72    1
10:  893025  -8930250  28279125  -30958200  15961050  -4396140  689850  -63000  3285  -90   1
-------------------------------------------------------------------------------------------------

R. P. Boas and R. C. Buck, Polynomial Expansions of Analytic Functions, Springer, 1958, p. 41
S. Roman, The Umbral Calculus, Academic Press, New York, 1984, p. 128.

G. C. Greubel, Rows n=0..100 of triangle, flattened

Row sums: A111882. Unsigned row sums: A111883.

Cf. A112239 (matrix log).

row[n_] := CoefficientList[ 1/2^n*HermiteH[n, Sqrt[x/2]]^2, x]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jul 17 2013 *)

from sympy import hermite, Poly, sqrt, symbols
x = symbols('x')
def a(n): return Poly(1/2**n*hermite(n, sqrt(x/2))**2, x).all_coeffs()[::-1]
for n in range(11): print(a(n)) # Indranil Ghosh, May 26 2017

E.g.f. for column m>=0: (1/sqrt(1-x^2))*((x/(1+x))^m)/m!.

a(n, m)=((-1)^(n-m))*(n!/m!)*sum(binomial(2*k, k)*binomial(n-2*k-1, m-1)/(4^k), k=0..floor((n-m)/2)), n>=m>=1. a(2*k, 0)= ((2*k)!/(k!*2^k))^2 = A001818(k), a(2*k+1) = 0, k>=0. a(n, m)=0 if n

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.

A060524 Triangle read by rows: T(n,k) = number of degree-n permutations with k odd cycles, k=0..n, n >= 0.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 5, 0, 1, 9, 0, 14, 0, 1, 0, 89, 0, 30, 0, 1, 225, 0, 439, 0, 55, 0, 1, 0, 3429, 0, 1519, 0, 91, 0, 1, 11025, 0, 24940, 0, 4214, 0, 140, 0, 1, 0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1, 893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1, 0

Offset: 0

Vladeta Jovovic, Apr 01 2001

The row polynomials t(n,x):=Sum_{k=0..n} T(n,k)*x^k satisfy the recurrence relation t(n,x) = x*t(n-1,x) + ((n-1)^2)*t(n-2,x); t(-1,x)=0, t(0,x)=1. - Wolfdieter Lang, see above.
This is an example of a Sheffer triangle (coefficient triangle for Sheffer polynomials). In the umbral calculus (see the Roman reference given under A048854) s(n,x) := Sum_{k=0..n} T(n,k)*x^k would be called Sheffer polynomials for (1/cosh(t),tanh(t)), which translates to the e.g.f. for column number k>=0 given by (1/sqrt(1-x^2))*((arctanh(x))^k)/k!. The e.g.f. given below is rewritten in this Sheffer context as (1/sqrt(1-x^2))*exp(y*log(sqrt((1+x)/(1-x))))= (1/sqrt(1-x^2))*exp(y*arctanh(x)). The rows of the Jabotinsky type triangle |A049218| provide the coefficients of the associated polynomials. - Wolfdieter Lang, Feb 24 2005
The solution of the differential-difference relation f(n+1,x)= (d/dx)f(n,x) + (n^2)*f(n-1,x), n >= 1, with inputs f(0,x) and f(1,x) = (d/dx)f(0,x) is f(n,x) = t(n,d_x)*f(0,x), with the differential operator d_x:=d/dx and the row polynomials t(n,x) defined above. This problem appears in a computation of thermo field dynamics where f(0,x)=1/cosh(x). See the triangle A060081. - Wolfdieter Lang, Feb 24 2005
The inverse of the Sheffer matrix T with elements T(n,k) is the Sheffer matrix A060081. - Wolfdieter Lang, Jul 22 2005
T(n,k)=0 if n-k= 1(mod 2), else T(n,k) = sum of M2(n,p), p from {1,...,A000041(n)} restricted to partitions with exactly k odd parts and any nonnegative number of even parts. For the M2-multinomial numbers in A-St order see A036039(n,p). - Wolfdieter Lang, Aug 07 2007

			Triangle begins:
  [1],
  [0, 1],
  [1, 0, 1],
  [0, 5, 0, 1],
  [9, 0, 14, 0, 1],
  [0, 89, 0, 30, 0, 1],
  [225, 0, 439, 0, 55, 0, 1],
  [0, 3429, 0, 1519, 0, 91, 0, 1],
  [11025, 0, 24940, 0, 4214, 0, 140, 0, 1],
  [0, 230481, 0, 122156, 0, 10038, 0, 204, 0, 1],
  [893025, 0, 2250621, 0, 463490, 0, 21378, 0, 285, 0, 1],
  [0, 23941125, 0, 14466221, 0, 1467290, 0, 41778, 0, 385, 0, 1],
  ...
Signed version begins:
  [1],
  [0, 1],
  [-1, 0, 1],
  [0, -5, 0, 1],
  [9, 0, -14, 0, 1],
  [0, 89, 0, -30, 0, 1],
  [-225, 0, 439, 0, -55, 0, 1],
  [0, -3429, 0, 1519, 0, -91, 0, 1],
  ...
From _Peter Bala_, Feb 23 2024: (Start)
Maple can verify the following series for Pi:
Row 1 polynomial R(1, x) = x:
Pi = 3 + 4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(1, 2*n)*R(1, 2*n+2)).
Row 3 polynomial R(3, x) = 5*x + x^3:
(3/2)^2 * Pi = 7 + 4*(3^4)*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(3, 2*n)*R(3, 2*n+2)).
Row 5 polynomial R(5, x) = 89*x + 30*x^3 + x^5:
((3*5)/(2*4))^2 * Pi = 11 + 4*(3*5)^4*Sum_{n >= 1} (-1)^(n+1)/((2*n + 1)*R(5, 2*n)*R(5, 2*n+2)). (End)

Alois P. Heinz, Rows n = 0..140, flattened
Marcelo Aguiar, Sarah Brauner, and Vic Reiner, Configuration spaces and peak representations, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 13. See p. 12.
Peter Bala, Meixner polynomials and Brouncker's continued fraction for Pi
Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
Adel Hamdi and Jiang Zeng, Orthogonal polynomials and operator orderings, J. Math. Phys., 51:043506, (2010); arXiv:1006.0808 [math.CO], 2010.
Eric Weisstein's World of Mathematics, Meixner polynomial of the first kind.

Cf. A060338, A060523, A094368, A028353 (col 1), A103916 (col 2), A103917 (col 3), A103918 (col 4).

Cf. A111594 (associated Sheffer polynomials), A142979, A142983.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(multinomial(n, n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1)*
      `if`(irem(i, 2)=1, x^j, 1), j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Mar 09 2015
# alternative
A060524 := proc(n,k)
    option remember;
    if nR. J. Mathar, Jul 06 2023

nn = 6; Range[0, nn]! CoefficientList[
   Series[(1 - x^2)^(-1/2) ((1 + x)/(1 - x))^(y/2), {x, 0, nn}], {x, y}] // Grid  (* Geoffrey Critzer, Aug 28 2012 *)

E.g.f.: (1+x)^((y-1)/2)/(1-x)^((y+1)/2).
T(n, k) = T(n-1, k-1) + ((n-1)^2)*T(n-2, k); T(-1, k):=0, T(n, -1):=0, T(0, 0)=1, T(n, k)=0 if nWolfdieter Lang, see above.
The Meixner polynomials defined by S_0(x)=1, S_1(x) = x; S_{n+1}(x) = x*S_n(x) - n^2*S_{n-1}(x) give a signed version of this triangle (cf. A060338). - N. J. A. Sloane, May 30 2013
From Peter Bala, Apr 10 2024: (Start)
The n-th row polynomial R(n, x) satisfies
(4*n + 2)*R(n, x) = (x + 1)*R(n, x+2) - (x - 1)*R(n, x-2).
Series for Pi involving the row polynomials R(n, x): for n >= 0 there holds
((2*n + 1)!!/(2^n*n!))^2 * Pi = (4*n + 3) + 4*((2*n + 1)!!^4) * Sum_{k >= 1} (-1)^(k+1)/((2*k + 1)*R(2*n+1, 2*k)*R(2*n+1, 2*k+2)). Cf. A142979 and A142983.
R(2*n, 0) = A001147(n)^2 = A001818(n); R(2*n+1, 0) = 0.
R(n, 1) = n! = A000142(n).
R(2*n, 2) = (4*n + 1)*A001147(n)^2 = (4*n + 1)*((2*n)!/(2^n*n!))^2;
R(2*n+1, 2) = 2*A001447(n+1)^2 = 2*(2*n + 1)!^2/(n!^2*4^n).
R(n, 3) = (2*n + 1)*n! = A007680(n). (End)

A111594 Triangle of arctanh numbers.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 24, 0, 20, 0, 1, 0, 0, 184, 0, 40, 0, 1, 0, 720, 0, 784, 0, 70, 0, 1, 0, 0, 8448, 0, 2464, 0, 112, 0, 1, 0, 40320, 0, 52352, 0, 6384, 0, 168, 0, 1, 0, 0, 648576, 0, 229760, 0, 14448, 0, 240, 0, 1

Offset: 0

Wolfdieter Lang, Aug 23 2005

Sheffer triangle associated to Sheffer triangle A060524.
For Sheffer triangles (matrices) see the explanation and S. Roman reference given under A048854.
The inverse matrix of A with elements a(n,m), n,m>=0, is given in A111593.
In the umbral calculus notation (see the S. Roman reference) this triangle would be called associated to (1,tanh(y)).
The row polynomials p(n,x):=sum(a(n,m)*x^m,m=0..n), together with the row polynomials s(n,x) of A060524 satisfy the exponential (or binomial) convolution identity s(n,x+y) = sum(binomial(n,k)*s(k,x)*p(n-k,y),k=0..n), n>=0.
Without the n=0 row and m=0 column and signed, this will become the Jabotinsky triangle A049218 (arctan numbers). For Jabotinsky matrices see the Knuth reference under A039692.
The row polynomials p(n,x) (defined above) have e.g.f. exp(x*arctanh(y)).
Exponential Riordan array [1, arctanh(x)] = [1, log(sqrt((1+x)/(1-x)))]. - Paul Barry, Apr 17 2008
Also the Bell transform of A005359. For the definition of the Bell transform see A264428. - Peter Luschny, Jan 27 2016

			Binomial convolution of row polynomials:
p(3,x)= 2*x+x^3; p(2,x)=x^2, p(1,x)= x, p(0,x)= 1,
together with those from A060524:
s(3,x)= 5*x+x^3; s(2,x)= 1+x^2, s(1,x)= x, s(0,x)= 1; therefore:
5*(x+y)+(x+y)^3 = s(3,x+y) = 1*s(0,x)*p(3,y) + 3*s(1,x)*p(2,y) + 3*s(2,x)*p(1,y) +1*s(3,x)*p(0,y) = 2*y+y^3 + 3*x*y^2 + 3*(1+x^2)*y + (5*x+x^3).
Triangle begins:
  1;
  0,   1;
  0,   0,    1;
  0,   2,    0,   1;
  0,   0,    8,   0,    1;
  0,  24,    0,  20,    0,  1;
  0,   0,  184,   0,   40,  0,   1;
  0, 720,    0, 784,    0, 70,   0, 1;
  0,   0, 8448,   0, 2464,  0, 112, 0, 1;
...

Wolfdieter Lang, First 10 rows.

Row sums: A000246.

Cf. A005359, A049218, A060524, A111593.

# The function BellMatrix is defined in A264428.
BellMatrix(n -> `if`(n::even, n!, 0), 10); # Peter Luschny, Jan 27 2016

rows = 10;
t = Table[If[EvenQ[n], n!, 0], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018, after Peter Luschny *)

# uses[riordan_array from A256893]
riordan_array(1, atanh(x), 9, exp=true) # Peter Luschny, Apr 19 2015

E.g.f. for column m>=0: ((arctanh(x))^m)/m!.
a(n, m) = coefficient of x^n of ((arctanh(x))^m)/m!, n>=m>=0, else 0.
a(n, m) = a(n-1, m-1) + (n-2)*(n-1)*a(n-2, m), a(n, -1):=0, a(0, 0)=1, a(n, m)=0 for n

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A001813 Quadruple factorial numbers: a(n) = (2n)!/n!.

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A091042 Triangle of even numbered entries of odd numbered rows of Pascal's triangle A007318.

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A196838 Numerators of coefficients of Bernoulli polynomials with rising powers of the variable.

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A049444 Generalized Stirling number triangle of first kind.

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A062137 Coefficient triangle of generalized Laguerre polynomials n!*L(n,3,x) (rising powers of x).

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