cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A293951 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type B_n joins.

Original entry on oeis.org

1, 9, 477, 74601, 25740261, 16591655817, 17929265150637, 30098784753112329, 74180579084559895221, 256937013876000351610089, 1208025937371403268201735037, 7494692521096248546330688437801, 59931935202159196095445595508666501
Offset: 0

Views

Author

N. J. A. Sloane, Oct 29 2017

Keywords

Links

Crossrefs

Cf. A293950, A293952, A293953.
Cf. A000182 (m=2), this seq (m=3), A273352 (m=4), A318258 (m=5).

Programs

  • Mathematica
    LMLlist[m_, len_] := Table[(-1)^(n + 1) (m n)!, {n, 1, len}]*
Delete [CoefficientList[Series[Log[MittagLefflerE[m, z]], {z, 0, len}], z], 1];
LMLlist[3, 13] (* Peter Luschny, Aug 26 2018 *)

Extensions

Terms a(11) and beyond from Lars Blomberg, Oct 31 2017

A318253 Coefficient of x of the OmegaPolynomials (A318146), T(n, k) = [x] P(n, k) with n>=1 and k>=0, square array read by ascending antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, -2, 0, 0, 1, -9, 16, 0, 0, 1, -34, 477, -272, 0, 0, 1, -125, 11056, -74601, 7936, 0, 0, 1, -461, 249250, -14873104, 25740261, -353792, 0, 0, 1, -1715, 5699149, -2886735625, 56814228736, -16591655817, 22368256, 0, 0, 1, -6434, 132908041, -574688719793, 122209131374375, -495812444583424, 17929265150637, -1903757312, 0
Offset: 1

Views

Author

Peter Luschny, Aug 22 2018

Keywords

Comments

Because in the case n=2 these numbers are the classical signed tangent numbers (A000182) we call T(n, k) also 'generalized tangent numbers' when studied in connection with generalized Bernoulli numbers.

Examples

			[n\k][0  1     2        3              4                   5  ...]
------------------------------------------------------------------
[1]   0, 1,    0,       0,             0,                  0, ...  [A063524]
[2]   0, 1,   -2,      16,          -272,               7936, ...  [A000182]
[3]   0, 1,   -9,     477,        -74601,           25740261, ...  [A293951]
[4]   0, 1,  -34,   11056,     -14873104,        56814228736, ...  [A273352]
[5]   0, 1, -125,  249250,   -2886735625,    122209131374375, ...  [A318258]
[6]   0, 1, -461, 5699149, -574688719793, 272692888959243481, ...
        [A010763]

Crossrefs

Cf. A318146, A181937, A063524, A000182, A293951, A273352, A318258.

Programs

  • Maple
    # Prints square array row-wise. The function OmegaPolynomial is defined in A318146.
for n from 1 to 6 do seq(coeff(OmegaPolynomial(n, k), x, 1), k=0..6) od;
# In the sequence format:
0, seq(seq(coeff(OmegaPolynomial(n-k+1, k), x, 1), k=0..n), n=1..9);
# Alternatively, based on the recurrence of the André numbers:
ANum := proc(m, n) option remember; if n = 0 then return 1 fi;
`if`(modp(n, m) = 0, -1, 1);  [seq(m*k, k=0..(n-1)/m)];
%%*add(binomial(n, k)*ANum(m, k), k in %) end:
TNum := proc(n,k) if k=1 then 1 elif k=0 or n=1 then 0 else ANum(n, n*k-1) fi end:
for n from 1 to 6 do seq(TNum(n, k), k = 0..6) od;
  • Mathematica
    OmegaPolynomial[m_, n_] := Module[{S}, S = Series[MittagLefflerE[m, z]^x, {z, 0, 10}]; Expand[(m*n)! Coefficient[S, z, n]]];
T[n_, k_] := D[OmegaPolynomial[n, k], x] /. x -> 0;
Table[T[n - k, k], {n, 1, 10}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, Nov 27 2023 *)
  • Sage
    # Prints the array row-wise. The function OmegaPolynomial is in A318146.
for m in (1..6):
    print([0] + [list(OmegaPolynomial(m, n))[1] for n in (1..6)])
# Alternatively, based on the recurrence of the André numbers:
@cached_function
def ANum(m, n):
    if n == 0: return 1
    t = [m*k for k in (0..(n-1)//m)]
    s = sum(binomial(n, k)*ANum(m, k) for k in t)
    return -s if m.divides(n) else s
def TNum(m, n):
    if n == 1: return 1
    if n == 0 or m == 1: return 0
    return ANum(m, m*n-1)
for m in (1..6): print([TNum(m, n) for n in (0..6)])

Formula

T(n, k) is the derivative of OmegaPolynomial(n, k) evaluated at x = 0.
Apart from the border cases n=1 and k=0 the generalized tangent numbers are a subset of the André numbers A181937; more precisely: T(n, k) is 1 if k = 1 else if k = 0 or n = 1 then T(n, k) = 0 else T(n,k) = (-1)^(n+1)*A181937(n, n*k-1).

A318258 a(n) = [x] OmegaPolynomial(5, n). OmegaPolynomials are defined in A318146.

Original entry on oeis.org

0, 1, -125, 249250, -2886735625, 122209131374375, -14455143383196875000, 4006210678487307667578125, -2297417123000769120910212890625, 2485076260705905645263720799941406250, -4719878705811419698488114573981055908203125
Offset: 0

Views

Author

Peter Luschny, Aug 22 2018

Keywords

Crossrefs

Cf. A318253 (case n=5), A318146.
Cf. A000182 (m=2), A293951 (m=3), A273352 (m=4), this seq (m=5).

Programs

  • Maple
    # The function OmegaPolynomial is defined in A318146.
seq(coeff(OmegaPolynomial(5, n), x, 1), n=0..11);
  • Mathematica
    LMlist[m_, len_] := Table[(m n)!, {n, 0, len}]*
CoefficientList[Series[Log[MittagLefflerE[m, z]], {z, 0, len}], z];
LMlist[5, 13]

A318148 Coefficients of the Omega polynomials of order 4, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -34, 35, 0, 11056, -16830, 5775, 0, -14873104, 27560780, -15315300, 2627625, 0, 56814228736, -119412815760, 84786627900, -24734209500, 2546168625, 0, -495812444583424, 1140896479608800, -948030209181000, 364143337057500, -65706427536750, 4509264634875
Offset: 0

Views

Author

Peter Luschny, Aug 22 2018

Keywords

Comments

The name 'Omega polynomial' is not a standard name.

Examples

			[0] [1]
[1] [0,           1]
[2] [0,         -34,            35]
[3] [0,       11056,        -16830,        5775]
[4] [0,   -14873104,      27560780,   -15315300,      2627625]
[5] [0, 56814228736, -119412815760, 84786627900, -24734209500, 2546168625]

Crossrefs

All row sums are 1, alternating row sums (taken absolute) are A211212.
T(n,1) ~ A273352(n), T(n,n) = A025036(n).
A023531 (m=1), A318146 (m=2), A318147 (m=3), this seq (m=4).

Programs

  • Maple
    # See A318146 for the missing functions.
FL([seq(CL(OmegaPolynomial(4, n)), n=0..8)]);
  • Mathematica
    (* OmegaPolynomials are defined in A318146 *)
Table[CoefficientList[OmegaPolynomial[4, n], x], {n, 0, 6}] // Flatten
  • Sage
    # See A318146 for the function OmegaPolynomial.
[list(OmegaPolynomial(4, n)) for n in (0..6)]

Formula

Omega(m, n, z) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m). We consider here the case m = 4 (for other cases see the cross-references).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.