A181985 -id:A181985 - cp's OEIS frontend

A211212 4-alternating permutations of length 4n.

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

A181991 n-alternating permutations of length 4n.

Original entry on oeis.org

1, 1385, 315523, 60376809, 11593285251, 2301250545971, 472105349529479, 99537885358650089, 21451428576293883859, 4705284467293276073635, 1047067375984978044542143, 235809039854522043890582835, 53644722291938408687646120103, 12309355014854205055828909176039

Offset: 1

Peter Luschny, Apr 05 2012

nonn

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

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. A181985, A030662, A211213.

A181991 := proc(n) local E, dim, i, k; dim := 4*n;
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; E[0, dim] end:
seq(A181991(n), n = 1..14);
# Alternatively:
a := (x) -> (4*x)!*(-1/(4*x)!+2/x!/(3*x)!+1/(2*x)!^2-3/x!^2/(2*x)!+1/x!^4):
seq(a(n), n=1..14); # Peter Luschny, Aug 13 2015

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[n, 4 + 1][[4 + 1]]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 17 2013, after Maple code in A181985 *)

a(n) = (4*n)!*(-1/(4*n)! + 2/(n!*(3*n)!) + 1/(2*n)!^2 - 3/(n!^2*(2*n)!) + 1/n!^4). - Peter Luschny, Aug 13 2015

A211213 n-alternating permutations of length 3n.

Original entry on oeis.org

1, 61, 1513, 33661, 750751, 17116009, 398840401, 9464040829, 227864057851, 5550936701311, 136526608389601, 3384729259165801, 84478081828015513, 2120572560190269841, 53494979095639780513, 1355345459896317255037, 34469858667289041256051, 879619727291950363099291

Offset: 1

Peter Luschny, Apr 05 2012

nonn

a(n) = A181985(n,3).

Peter Luschny, An old operation on sequences: the Seidel transform

Cf. A181985, A030662, A181991.

A211213 := proc(n) local E, dim, i, k; dim := 3*n;
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; E[0, dim] end:
seq(A211213(n), n = 1..18);
# Alternatively:
a := x -> (3*x)!*(1/(3*x)!-2/(x!*(2*x)!)+1/(x!)^3):
seq(a(n),n=1..18); # Peter Luschny, Aug 13 2015

nmax = 18; a[n_] := Module[{e, dim = n*(nmax-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] ] ]]; e[0, 3*n]] ; Table[a[n], {n, 1, nmax}] (* Jean-François Alcover, Jul 26 2013, after Maple *)

a(n) = (3*n)!*(1/(3*n)!-2/(n!*(2*n)!)+1/(n!)^3). - Peter Luschny, Aug 13 2015

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.

A292606 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).

Original entry on oeis.org

1, 1, 0, 69, 1, 0, 33661, 988, 1, 0, 60376809, 2669683, 16507, 1, 0, 288294050521, 17033188586, 212734266, 261626, 1, 0, 3019098162602349, 223257353561605, 4297382231090, 17634518610, 4196345, 1, 0

Offset: 0

Peter Luschny, Sep 26 2017

See the comments in A292604.

			Triangle starts:
[n\k][          0            1          2       3   4   5]
--------------------------------------------------
[0] [           1]
[1] [           1,           0]
[2] [          69,           1,         0]
[3] [       33661,         988,         1,      0]
[4] [    60376809,     2669683,     16507,      1,  0]
[5] [288294050521, 17033188586, 212734266, 261626,  1,  0]

F_{0} = A129186, F_{1} = A173018, F_{2} = A292604, F_{3} = A292605, F_{4} is this triangle.
First column: A211212. Row sums: A014608. Alternating row sums: A292607.
Cf. A181985.

Coeffs := f -> PolynomialTools:-CoefficientList(expand(f), x):
A292606_row := proc(n) if n = 0 then return [1] fi;
add(A278074(n, k)*(x-1)^(n-k), k=0..n); [op(Coeffs(%)), 0] end:
for n from 0 to 6 do A292606_row(n) od;

# uses[A278074_row from A278074]
def A292606_row(n):
    if n == 0: return [1]
    L = A278074_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(A292606_row(n))

F_{4; n}(x) = Sum_{k=0..n} A278074(n, k)*(x-1)^(n-k) for n>0 and F_{4; 0}(x) = 1.

A181992 n-alternating permutations of length n^2.

Original entry on oeis.org

1, 1, 5, 1513, 60376809, 613498040952501, 2655748106132754540814283, 7350748555338515554166266981278924209, 18155845241010181420704703186769135339279915667193169, 53121946985233865823079732996510797894348260342024814486694637630897821

Offset: 0

Peter Luschny, Apr 05 2012

nonn

These are the generalized Euler numbers A181985(n, n) and also the André numbers A181937(n, n^2).

Alois P. Heinz, Table of n, a(n) for n = 0..26
Peter Luschny, An old operation on sequences: the Seidel transform

Cf. A181985, A181937.

A181992 := proc(n) local E, dim, i, k; dim := n*n;
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;
E[0, dim] end:
seq(A181992(i),i=0..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}]]; a[n_] := A181985[n, n + 1][[n + 1]]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Dec 17 2013, after Maple code in A181985 *)

a(0)=1 prepended by Alois P. Heinz, Aug 12 2019

cp's OEIS Frontend

A211212 4-alternating permutations of length 4n.

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A181991 n-alternating permutations of length 4n.

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A211213 n-alternating permutations of length 3n.

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

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

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A292606 Triangle read by rows, coefficients of generalized Eulerian polynomials F_{4;n}(x).

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A181992 n-alternating permutations of length n^2.

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