A002115 -id:A002115 - cp's OEIS frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

1, 0, 1, 0, 1, 20, 0, 1, 168, 1680, 0, 1, 1364, 55440, 369600, 0, 1, 10920, 1561560, 33633600, 168168000, 0, 1, 87380, 42771456, 2385102720, 34306272000, 137225088000, 0, 1, 699048, 1160164320, 158411809920, 5105916816000, 54752810112000, 182509367040000

Offset: 0

Peter Luschny, Jan 22 2017

			Triangle begins:
[1]
[0, 1]
[0, 1,    20]
[0, 1,   168,    1680]
[0, 1,  1364,   55440,   369600]
[0, 1, 10920, 1561560, 33633600, 168168000]

Cf. A014606 (diagonal), A243664 (row sums), A002115 (alternating row sums), A281479 (central coefficients), A327023 (refinement).

Cf. A097805 (m=0), A131689 (m=1), A241171 (m=2), A278074 (m=4).

P := proc(m, n) option remember; if n = 0 then 1 else
add(binomial(m*n, m*k)*P(m, n-k)*x, k=1..n) fi end:
for n from 0 to 6 do PolynomialTools:-CoefficientList(P(3,n), x) od;
# Alternatively:
A278073_row := proc(n)
1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1));
expand(series(%,x,3*n+1)); (3*n)!*coeff(%,x,3*n);
PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 6 do A278073_row(n) od;

With[{m = 3}, Table[Expand[j!*SeriesCoefficient[1/(1 - t*(MittagLefflerE[m, x^m] - 1)), {x, 0, j}]], {j, 0, 21, m}]];
Function[arg, CoefficientList[arg, t]] /@ % // Flatten

R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(30)
@cached_function
def P(m, n):
    if n == 0: return R(1)
    return expand(sum(binomial(m*n, m*k)*P(m, n-k)*x for k in (1..n)))
def A278073_row(n): return list(P(3, n))
for n in (0..6): print(A278073_row(n)) # Peter Luschny, Mar 24 2020

E.g.f.: 1/(1-t*((1/3)*exp(x)+(2/3)*exp(-(1/2)*x)*cos((1/2)*x*sqrt(3))-1)), nonzero terms.

A178963 E.g.f.: (3+2sqrt(3)exp(x/2)sin(sqrt(3)x/2))/(exp(-x)+2exp(x/2)cos(sqrt(3)*x/2)).

Original entry on oeis.org

1, 1, 1, 1, 3, 9, 19, 99, 477, 1513, 11259, 74601, 315523, 3052323, 25740261, 136085041, 1620265923, 16591655817, 105261234643, 1488257158851, 17929265150637, 132705221399353, 2172534146099019, 30098784753112329, 254604707462013571, 4736552519729393091, 74180579084559895221, 705927677520644167681, 14708695606607601165843, 256937013876000351610089, 2716778010767155313771539

Offset: 0

N. J. A. Sloane, Dec 31 2010

nonn

According to Mendes and Remmel, p. 56, this is the e.g.f. for 3-alternating permutations.

Alois P. Heinz, Table of n, a(n) for n = 0..500
J. M. Luck, On the frequencies of patterns of rises and falls, arXiv preprint arXiv:1309.7764 [cond-mat.stat-mech], 2013-2014.
Peter Luschny, An old operation on sequences: the Seidel transform.
Anthony Mendes and Jeffrey Remmel, Generating functions from symmetric functions, Preliminary version of book, available from Jeffrey Remmel's home page.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Number of m-alternating permutations: A000012 (m=1), A000111 (m=2), A178963 (m=3), A178964 (m=4), A181936 (m=5).
Cf. A181937, A002115.
Cf. A249402, A249583 (alternative definitions of 3-alternating permutations).

A178963_list := proc(dim) local E,DIM,n,k;
DIM := dim-1; E := array(0..DIM, 0..DIM); E[0,0] := 1;
for n from 1 to DIM do
if n mod 3 = 0 then E[n,0] := 0 ;
   for k from n-1 by -1 to 0 do E[k,n-k] := E[k+1,n-k-1] + E[k,n-k-1] od;
else E[0,n] := 0;
   for k from 1 by 1 to n do E[k,n-k] := E[k-1,n-k+1] + E[k-1,n-k] od;
fi od; [E[0,0],seq(E[k,0]+E[0,k],k=1..DIM)] end:
A178963_list(30);  # Peter Luschny, Apr 02 2012
# Alternatively, using a bivariate exponential generating function:
A178963 := proc(n) local g, p, q;
g := (x,z) -> 3*exp(x*z)/(exp(z)+2*exp(-z/2)*cos(z*sqrt(3)/2));
p := (n,x) -> n!*coeff(series(g(x,z),z,n+2),z,n);
q := (n,m) -> if modp(n,m) = 0 then 0 else 1 fi:
(-1)^floor(n/3)*p(n,q(n,3)) end:
seq(A178963(i),i=0..30); # Peter Luschny, Jun 06 2012
# third Maple program:
b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
     `if`(t=0, add(b(u-j, o+j-1, irem(t+1, 3)), j=1..u),
               add(b(u+j-1, o-j, irem(t+1, 3)), j=1..o)))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..35);  # Alois P. Heinz, Oct 29 2014

max = 30; f[x_] := (E^x*(2*Sqrt[3]*E^(x/2)*Sin[(Sqrt[3]*x)/2] + 3))/(2*E^((3*x)/2)*Cos[(Sqrt[3]*x)/2] + 1); CoefficientList[Series[f[x], {x, 0, max}], x]*Range[0, max]! // Simplify (* Jean-François Alcover, Sep 16 2013 *)

# uses[A from A181936]
A178963 = lambda n: (-1)^int(is_odd(n//3))*A(3,n)
print([A178963(n) for n in (0..30)]) # Peter Luschny, Jan 24 2017

a(3*n) = A002115(n). - Peter Luschny, Aug 02 2012

A181985 Generalized Euler numbers. Square array A(n,k), n >= 1, k >= 0, read by antidiagonals. A(n,k) = n-alternating permutations of length n*k.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 19, 61, 1, 1, 1, 69, 1513, 1385, 1, 1, 1, 251, 33661, 315523, 50521, 1, 1, 1, 923, 750751, 60376809, 136085041, 2702765, 1, 1, 1, 3431, 17116009, 11593285251, 288294050521, 105261234643, 199360981, 1

Offset: 1

Peter Luschny, Apr 04 2012

For an integer n > 0, a permutation s = s_1...s_k is an n-alternating permutation if it has the property that s_i < s_{i+1} if and only if n divides i.
The classical Euler numbers count 2-alternating permutations of length 2n.
Ludwig Seidel gave in 1877 an efficient algorithm to compute the coefficients of sec which carries immediately over to the computation of the generalized Euler numbers (see the Maple script).

			n\k [0][1]  [2]       [3]            [4]                 [5]
[1]  1, 1,   1,        1,             1,                   1
[2]  1, 1,   5,       61,          1385,               50521  [A000364]
[3]  1, 1,  19,     1513,        315523,           136085041  [A002115]
[4]  1, 1,  69,    33661,      60376809,        288294050521  [A211212]
[5]  1, 1, 251,   750751,   11593285251,     613498040952501
[6]  1, 1, 923, 17116009, 2301250545971, 1364944703949044401
       [A030662][A211213]   [A181991]
The (n,n)-diagonal is A181992.

Peter Luschny, An old operation on sequences: the Seidel transform.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]

Cf. A181937, A000364, A002115, A030662, A211212, A211213, A181991, A181992.

A181985_list := proc(n, len) local E, dim, i, k;
dim := n*(len-1); E := array(0..dim, 0..dim); E[0, 0] := 1;
for i from 1 to dim do
   if i mod n = 0 then E[i, 0] := 0 ;
      for k from i-1 by -1 to 0 do E[k, i-k] := E[k+1, i-k-1] + E[k, i-k-1] od;
   else E[0, i] := 0;
      for k from 1 by 1 to i do E[k, i-k] := E[k-1, i-k+1] + E[k-1, i-k] od;
   fi od;
seq(E[0, n*k], k=0..len-1) end:
for n from 1 to 6 do print(A181985_list(n, 6)) od;

nmax = 9; A181985[n_, len_] := Module[{e, dim = n*(len - 1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0 , e[i, 0] = 0 ; For[k = i-1, k >= 0, k--, e[k, i-k] = e[k+1, i-k-1] + e[k, i-k-1] ], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i-k] = e[k-1, i-k+1] + e[k-1, i-k] ]; ]]; Table[e[0, n*k], { k, 0, len-1}]]; t = Table[A181985[n, nmax], {n, 1, nmax}]; a[n_, k_] := t[[n, k+1]]; Table[a[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jun 27 2013, translated and adapted from Maple *)

def A181985(m, n):
    shapes = ([x*m for x in p] for p in Partitions(n))
    return (-1)^n*sum((-1)^len(s)*factorial(len(s))*SetPartitions(sum(s), s).cardinality() for s in shapes)
for m in (1..6): print([A181985(m, n) for n in (0..7)]) # Peter Luschny, Aug 10 2015

A211212 4-alternating permutations of length 4n.

Original entry on oeis.org

1, 1, 69, 33661, 60376809, 288294050521, 3019098162602349, 60921822444067346581, 2159058013333667522020689, 125339574046311949415000577841, 11289082167259099068433198467575829, 1510335441937894173173702826484473600301

Offset: 0

Peter Luschny, Apr 04 2012

nonn

a(n) = A181985(4,n).

L. Carlitz, Permutations with prescribed pattern, Math. Nachr. 58 (1973), 31-53.
Peter Luschny, An old operation on sequences: the Seidel transform

Cf. A000364, A002115, A181985.

A211212 := proc(n) local E, dim, i, k; dim := 4*(n-1);
E := array(0..dim, 0..dim); E[0, 0] := 1;
for i from 1 to dim do
   if i mod 4 = 0 then E[i, 0] := 0 ;
      for k from i-1 by -1 to 0 do E[k, i-k] := E[k+1, i-k-1] + E[k, i-k-1] od;
   else E[0, i] := 0;
      for k from 1 by 1 to i do E[k, i-k] := E[k-1, i-k+1] + E[k-1, i-k] od;
   fi od;
E[0, dim] end:
seq(A211212(i), i = 1..12);
A211212_list := proc(size) local E, S;
E := 2*exp(x*z)/(cosh(z)+cos(z));
S := z -> series(E, z, 4*(size+1));
seq((-1)^n*(4*n)!*subs(x=0, coeff(S(z), z, 4*n)), n=0..size-1) end:
A211212_list(12); # Peter Luschny, Jun 06 2016

A181985[n_, len_] := Module[{e, dim = n (len - 1)}, e[0, 0] = 1; For[i = 1, i <= dim, i++, If[Mod[i, n] == 0, e[i, 0] = 0; For[k = i - 1, k >= 0, k--, e[k, i - k] = e[k + 1, i - k - 1] + e[k, i - k - 1]], e[0, i] = 0; For[k = 1, k <= i, k++, e[k, i - k] = e[k - 1, i - k + 1] + e[k - 1, i - k]]]]; Table[e[0, n k], {k, 0, len - 1}]];
a[n_] := A181985[4, n + 1] // Last;
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Jun 29 2019 *)

# uses[A from A181936]
A211212 = lambda n: A(4,4*n)*(-1)^n
print([A211212(n) for n in (0..11)]) # Peter Luschny, Jan 24 2017

a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k+1) * binomial(4*n,4*k) * a(n-k). - Ilya Gutkovskiy, Jan 27 2020
E.g.f.: 1/(cos(x/sqrt(2))*cosh(x/sqrt(2))) = 1 + 1*z^4/4! + 69*z^8/8! + 33661*z^12/12! + ... - Michael Wallner, Nov 17 2020
a(n) ~ 2^(10*n + 9/2) * n^(4*n + 1/2) / (cosh(Pi/2) * Pi^(4*n + 1/2) * exp(4*n)). - Vaclav Kotesovec, Nov 17 2020

A215064 Triangle read by rows, e.g.f. exp(xz)((exp(x/2)+exp(x3/2))/((exp(3x/2)+ 2cos(sqrt(3)x/2))/3)-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, -1, 3, 3, 1, -3, -4, 6, 4, 1, -9, -15, -10, 10, 5, 1, 19, -54, -45, -20, 15, 6, 1, 99, 133, -189, -105, -35, 21, 7, 1, 477, 792, 532, -504, -210, -56, 28, 8, 1, -1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1, -11259

Offset: 0

Peter Luschny, Aug 01 2012

			[0] [1]
[1] [1, 1]
[2] [1, 2, 1]
[3] [-1, 3, 3, 1]
[4] [-3, -4, 6, 4, 1]
[5] [-9, -15, -10, 10, 5, 1]
[6] [19, -54, -45, -20, 15, 6, 1]
[7] [99, 133, -189, -105, -35, 21, 7, 1]
[8] [477, 792, 532, -504, -210, -56, 28, 8, 1]
[9] [-1513, 4293, 3564, 1596, -1134, -378, -84, 36, 9, 1]

Cf. A215060, A215061, A215062, A215063, A215065.

max = 11; f = Exp[x*z]*((Exp[x/2] + Exp[x*(3/2)])/((Exp[3*(x/2)] + 2*Cos[Sqrt[3]*(x/2)])/3) - 1); coes = CoefficientList[ Series[f, {x, 0, max}, {z, 0, max}], {x, z}]; Table[ coes[[n, k]]*(n - 1)!, {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

# uses[triangle from A215060]
def A215064_triangle(dim):
    var('x, z')
    f = exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)-1)
    return triangle(f, dim)
A215064_triangle(12)

Matrix inverse is A215065.
T(n,k) = A215060(n,k) + A215062(n,k) - [n==k].
|T(n,0)| = A178963(n).
|T(3*n,0)| = A002115(n).

A215060 Triangle read by rows, e.g.f. exp(x(z+1/2))/((exp(3x/2) + 2cos(sqrt(3)x/2))/3).

Original entry on oeis.org

1, 0, 1, 0, 0, 1, -1, 0, 0, 1, 0, -4, 0, 0, 1, 0, 0, -10, 0, 0, 1, 19, 0, 0, -20, 0, 0, 1, 0, 133, 0, 0, -35, 0, 0, 1, 0, 0, 532, 0, 0, -56, 0, 0, 1, -1513, 0, 0, 1596, 0, 0, -84, 0, 0, 1, 0, -15130, 0, 0, 3990, 0, 0, -120, 0, 0, 1, 0, 0, -83215, 0

Offset: 0

Peter Luschny, Aug 01 2012

			[0] [1]
[1] [0, 1]
[2] [0, 0, 1]
[3] [-1, 0, 0, 1]
[4] [0, -4, 0, 0, 1]
[5] [0, 0, -10, 0, 0, 1]
[6] [19, 0, 0, -20, 0, 0, 1]
[7] [0, 133, 0, 0, -35, 0, 0, 1]
[8] [0, 0, 532, 0, 0, -56, 0, 0, 1]
[9] [-1513, 0, 0, 1596, 0, 0, -84, 0, 0, 1]

Cf. A215061, A215062, A215063, A215064, A215065.

def triangle(f, dim):
    var('x,z')
    s = f.series(x, dim+2)
    P = [factorial(i)*s.coefficient(x,i) for i in range(dim)]
    for k in range(dim): print([k], [P[k].coefficient(z,i) for i in (0..k)])
def A215060_triangle(dim) :
    var('x, z')
    f = exp(x*(z+1/2))/((exp(3*x/2)+2*cos(sqrt(3)*x/2))/3)
    return triangle(f, dim)
A215060_triangle(12)

Matrix inverse is A215061.
T(n,k) = A215064(n,k) - A215062(n,k) + [n==k].
|T(3*n,0)| = A002115(n).

A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

Original entry on oeis.org

1, 1, 0, 5, 1, 0, 61, 28, 1, 0, 1385, 1011, 123, 1, 0, 50521, 50666, 11706, 506, 1, 0, 2702765, 3448901, 1212146, 118546, 2041, 1, 0, 199360981, 308869464, 147485535, 24226000, 1130235, 8184, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

The generalized Eulerian polynomials F_{m}(x) are defined F_{m; 0}(x) = 1 for all m >= 0 and for n > 0:
F_{0; n}(x) = Sum_{k=0..n} A097805(n, k)*(x-1)^(n-k) with coeffs. in A129186.
F_{1; n}(x) = Sum_{k=0..n} A131689(n, k)*(x-1)^(n-k) with coeffs. in A173018.
F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) with coeffs. in A292604.
F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) with coeffs. in A292605.
F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) with coeffs. in A292606.
The case m = 1 are the Eulerian polynomials whose coefficients are the Eulerian numbers which are displayed in Euler's triangle A173018.
Evaluated at x in {-1, 1, 0} these families of polynomials give for the first few m:
F_{m} :   F_{0}    F_{1}    F_{2}    F_{3}    F_{4}
x = -1:  A165326  A155585  A002105  A292609  A292607
x =  1:  A000012  A000142  A000680  A014606  A014608  ... (m*n)!/m!^n
x =  0:    --     A000012  A000364  A002115  A211212  ... m-alternating permutations of length m*n.
Note that the constant terms of the polynomials are the generalized Euler numbers as defined in A181985. In this sense generalized Euler numbers are also generalized Eulerian numbers.

			Triangle starts:
[n\k][    0        1        2       3     4  5  6]
--------------------------------------------------
[0][      1]
[1][      1,       0]
[2][      5,       1,       0]
[3][     61,      28,       1,      0]
[4][   1385,    1011,     123,      1,    0]
[5][  50521,   50666,   11706,    506,    1, 0]
[6][2702765, 3448901, 1212146, 118546, 2041, 1, 0]

G. Frobenius. Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber. Preuss. Akad. Wiss. Berlin, pages 200-208, 1910.

F_{0} = A129186, F_{1} = A173018, F_{2} is this triangle, F_{3} = A292605, F_{4} = A292606.
First column: A000364. Row sums: A000680. Alternating row sums: A002105.
Cf. A181985, A241171.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292604_row := proc(n) if n = 0 then return [1] fi;
add(A241171(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292604_row(n) od;

T[n_, k_] /; 1 <= k <= n := T[n, k] = k (2 k - 1) T[n - 1, k - 1] + k^2 T[n - 1, k]; T[, 1] = 1; T[, _] = 0;
F[2, 0][] = 1; F[2, n][x_] := Sum[T[n, k] (x - 1)^(n - k), {k, 0, n}];
row[n_] := If[n == 0, {1}, Append[CoefficientList[ F[2, n][x], x], 0]];
Table[row[n], {n, 0, 7}] (* Jean-François Alcover, Jul 06 2019 *)

def A292604_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..6): print(A292604_row(n))

F_{2; n}(x) = Sum_{k=0..n} A241171(n, k)*(x-1)^(n-k) for n>0 and F_{2; 0}(x) = 1.

A292605 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{3;n}(x).

Original entry on oeis.org

1, 1, 0, 19, 1, 0, 1513, 166, 1, 0, 315523, 52715, 1361, 1, 0, 136085041, 30543236, 1528806, 10916, 1, 0, 105261234643, 29664031413, 2257312622, 42421946, 87375, 1, 0, 132705221399353, 45011574747714, 4637635381695, 153778143100, 1156669095, 699042, 1, 0

Offset: 0

Peter Luschny, Sep 20 2017

See the comments in A292604.

			Triangle starts:
[n\k][       0         1         2       3  4  5]
--------------------------------------------------
[0][         1]
[1][         1,        0]
[2][        19,        1,        0]
[3][      1513,      166,        1,     0]
[4][    315523,    52715,     1361,     1,  0]
[5][ 136085041, 30543236,  1528806, 10916,  1, 0]

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} is this triangle, F_{4} = A292606.
First column: A002115. Row sums: A014606. Alternating row sums: A292609.
Cf. A181985, A278073.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f),x):
A292605_row := proc(n) if n = 0 then return [1] fi;
add(A278073(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292605_row(n) od;

# uses[A278073_row from A278073]
def A292605_row(n):
    if n == 0: return [1]
    L = A278073_row(n)
    S = sum(L[k]*(x-1)^(n-k) for k in (0..n))
    return expand(S).list() + [0]
for n in (0..5): print(A292605_row(n))

F_{3; n}(x) = Sum_{k=0..n} A278073(n, k)*(x-1)^(n-k) for n>0 and F_{3; 0}(x) = 1.

A326475 A(n, k) = (m*k)! [x^k] MittagLefflerE(m, x)^(-n), for m = 3, n >= 0, k >= 0; square array read by descending antidiagonals.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, 19, -2, 1, 0, -1513, 58, -3, 1, 0, 315523, -6218, 117, -4, 1, 0, -136085041, 1630330, -15795, 196, -5, 1, 0, 105261234643, -847053482, 4997781, -31924, 295, -6, 1, 0, -132705221399353, 766492673914, -3042574083, 11840836, -56285, 414, -7, 1

Offset: 0

Peter Luschny, Jul 08 2019

			Array starts:
[0] 1,  0,   0,      0,        0,            0, ... A000007
[1] 1, -1,  19,  -1513,   315523,   -136085041, ... A002115
[2] 1, -2,  58,  -6218,  1630330,   -847053482, ...
[3] 1, -3, 117, -15795,  4997781,  -3042574083, ...
[4] 1, -4, 196, -31924, 11840836,  -8271354004, ...
[5] 1, -5, 295, -56285, 23952055, -18889306805, ...
[6] 1, -6, 414, -90558, 43493598, -38227720446, ...

Rows: A000007, A002115.

Cf. A326476 (m=2, p>=0), A326327 (m=2, p<=0), A326474 (m=3, p>=0), this sequence (m=3, p<=0).

(* The function MLPower is defined in A326327. *)
For[n = 0, n < 8, n++, Print[MLPower[3, -n, 8]]]

# uses[MLPower from A326327]
for n in (0..6): print(MLPower(3, -n, 9))

A327023 Ordered set partitions of the set {1, 2, ..., 3*n} with all block sizes divisible by 3, irregular triangle T(n, k) for n >= 0 and 0 <= k < A000041(n), read by rows.

Original entry on oeis.org

1, 1, 1, 20, 1, 168, 1680, 1, 440, 924, 55440, 369600, 1, 910, 10010, 300300, 1261260, 33633600, 168168000, 1, 1632, 37128, 48620, 1113840, 24504480, 17153136, 326726400, 2058376320, 34306272000, 137225088000

Offset: 0

Peter Luschny, Aug 27 2019

T_{m}(n, k) gives the number of ordered set partitions of the set {1, 2, ..., m*n} into sized blocks of shape m*P(n, k), where P(n, k) is the k-th integer partition of n in the 'canonical' order A080577. Here we assume the rows of A080577 to be 0-based and m*[a, b, c,..., h] = [m*a, m*b, m*c,..., m*h]. Here is case m = 3. For instance 3*P(4, .) = [[12], [9, 3], [6, 6], [6, 3, 3], [3, 3, 3, 3]].

			Triangle starts (note the subdivisions by ';' (A072233)):
[0] [1]
[1] [1]
[2] [1;   20]
[3] [1;  168;  1680]
[4] [1;  440,   924;  55440;  369600]
[5] [1;  910, 10010; 300300, 1261260; 33633600; 168168000]
[6] [1; 1632, 37128,  48620; 1113840, 24504480,  17153136; 326726400, 2058376320;
     34306272000; 137225088000]
.
T(4, 1) = 440 because [9, 3] is the integer partition 3*P(4, 1) in the canonical order and there are 220 set partitions which have the shape [9, 3]. Finally, since the order of the sets is taken into account, one gets 2!*220 = 440.

Row sums: A243664, alternating row sums: A002115, main diagonal: A014606, central column A281479, by length: A278073.
Cf. A178803 (m=0), A133314 (m=1), A327022 (m=2), this sequence (m=3), A327024 (m=4).
Cf. A080577, A000041, A072233.

# uses[GenOrdSetPart from A327022]
def A327023row(n): return GenOrdSetPart(3, n)
for n in (0..6): print(A327023row(n))

cp's OEIS Frontend

A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(m*n, m*k)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

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A178963 E.g.f.: (3+2*sqrt(3)*exp(x/2)*sin(sqrt(3)*x/2))/(exp(-x)+2*exp(x/2)*cos(sqrt(3)*x/2)).

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A181985 Generalized Euler numbers. Square array A(n,k), n >= 1, k >= 0, read by antidiagonals. A(n,k) = n-alternating permutations of length n*k.

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A211212 4-alternating permutations of length 4n.

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A215064 Triangle read by rows, e.g.f. exp(x*z)*((exp(x/2)+exp(x*3/2))/((exp(3*x/2)+ 2*cos(sqrt(3)*x/2))/3)-1).

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A215060 Triangle read by rows, e.g.f. exp(x*(z+1/2))/((exp(3*x/2) + 2*cos(sqrt(3)*x/2))/3).

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A292604 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{2}(x).

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A278073 Triangle read by rows, coefficients of the polynomials P(m, n) = Sum_{k=1..n} binomial(mn, mk)* P(m, n-k)*z with P(m, 0) = 1 and m = 3.

A178963 E.g.f.: (3+2sqrt(3)exp(x/2)sin(sqrt(3)x/2))/(exp(-x)+2exp(x/2)cos(sqrt(3)*x/2)).

A215064 Triangle read by rows, e.g.f. exp(xz)((exp(x/2)+exp(x3/2))/((exp(3x/2)+ 2cos(sqrt(3)x/2))/3)-1).

A215060 Triangle read by rows, e.g.f. exp(x(z+1/2))/((exp(3x/2) + 2cos(sqrt(3)x/2))/3).