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 1-3 of 3 results.

A318146 Coefficients of the Omega polynomials of order 2, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -2, 3, 0, 16, -30, 15, 0, -272, 588, -420, 105, 0, 7936, -18960, 16380, -6300, 945, 0, -353792, 911328, -893640, 429660, -103950, 10395, 0, 22368256, -61152000, 65825760, -36636600, 11351340, -1891890, 135135
Offset: 0

Views

Author

Peter Luschny, Aug 22 2018

Keywords

Comments

The name 'Omega polynomial' is not a standard name. The Omega numbers are the coefficients of the Omega polynomials, the associated Omega numbers are the weights of P(m, k) in the recurrence formula given below.
The signed Euler secant numbers appear as values at x=-1 and the signed Euler tangent numbers as the coefficients of x.

Examples

			Row n in the triangle below is the coefficient list of OmegaPolynomial(2, n). For other cases than m = 2 see the cross-references.
[0] [1]
[1] [0,        1]
[2] [0,       -2,         3]
[3] [0,       16,       -30,       15]
[4] [0,     -272,       588,     -420,       105]
[5] [0,     7936,    -18960,    16380,     -6300,      945]
[6] [0,  -353792,    911328,  -893640,    429660,  -103950,    10395]
[7] [0, 22368256, -61152000, 65825760, -36636600, 11351340, -1891890, 135135]

Links

Crossrefs

Variant is A088874 (unsigned).
T(n,1) = A000182(n), T(n,n) = A001147(n).
All row sums are 1, alternating row sums are A028296 (A000364).
A023531 (m=1), this seq (m=2), A318147 (m=3), A318148 (m=4).
Associated Omega numbers: A318254 (m=2), A318255 (m=3).
Coefficients of x for Omega polynomials of all orders are in A318253.

Programs

  • Maple
    OmegaPolynomial := proc(m, n) local Omega;
Omega := m -> hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m):
series(Omega(m)^x, z, m*(n+1)):
sort(expand((m*n)!*coeff(%, z, n*m)), [x], ascending) end:
CL := p -> PolynomialTools:-CoefficientList(p, x):
FL := p -> ListTools:-Flatten(p):
FL([seq(CL(OmegaPolynomial(2, n)), n=0..8)]);
# Alternative:
ser := series(sech(z)^(-x), z, 24): row := n -> n!*coeff(ser, z, n):
seq(seq(coeff(row(2*n), x, k), k=0..n), n=0..6); # Peter Luschny, Jul 01 2019
  • Mathematica
    OmegaPolynomial[m_,n_] :=  Module [{ },
S = Series[MittagLefflerE[m,z]^x, {z,0,10}];
Expand[(m n)! Coefficient[S,z,n]] ]
Table[CoefficientList[OmegaPolynomial[2,n],x], {n,0,7}] // Flatten
(* Second program: *)
T[n_, k_] := (2n)! SeriesCoefficient[Sech[z]^-x, {z, 0, 2n}, {x, 0, k}];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 23 2019, after Peter Luschny *)
  • Sage
    def OmegaPolynomial(m, n):
    R = ZZ[x]; z = var('z')
    f = [i/m for i in (1..m-1)]
    h = lambda z: hypergeometric([], f, (z/m)^m)
    return R(factorial(m*n)*taylor(h(z)^x, z, 0, m*n + 1).coefficient(z, m*n))
[list(OmegaPolynomial(2, n)) for n in (0..6)]
# Recursion over the polynomials, returns a list of the first len polynomials:
def OmegaPolynomials(m, len, coeffs=true):
    R = ZZ[x]; B = [0]*len; L = [R(1)]*len
    for k in (1..len-1):
        s = x*sum(binomial(m*k-1, m*(k-j))*B[j]*L[k-j] for j in (1..k-1))
        B[k] = c = 1 - s.subs(x=1)
        L[k] = R(expand(s + c*x))
    return [list(l) for l in L] if coeffs else L
print(OmegaPolynomials(2, 6))

Formula

OmegaPolynomial(m, n) = (m*n)! [z^n] E(m, z)^x where E(m, z) is the Mittag-Leffler function.
OmegaPolynomial(m, n) = (m*n)!*[z^(n*m)] H(m, z)^x where H(m, z) = hypergeom([], [seq(i/m, i=1..m-1)], (z/m)^m).
The Omega polynomials can be computed by the recurrence P(m, 0) = 1 and for n >= 1 P(m, n) = x * Sum_{k=0..n-1} binomial(m*n-1, m*k)*T(m, n-k)*P(m, k) where T(m, n) are the generalized tangent numbers A318253. A separate computation of the T(m, n) can be avoided, see the Sage implementation below for the details.
T(n, k) = [x^k] (2*n)! [z^(2*n)] sech(z)^(-x). - Peter Luschny, Jul 01 2019

A363393 Triangle read by rows. T(n, k) = [x^k] (2 - Sum_{k=0..n} binomial(n, k)*Euler(k, 1)*(-2*x)^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 3, 0, -2, 1, 4, 0, -8, 0, 1, 5, 0, -20, 0, 16, 1, 6, 0, -40, 0, 96, 0, 1, 7, 0, -70, 0, 336, 0, -272, 1, 8, 0, -112, 0, 896, 0, -2176, 0, 1, 9, 0, -168, 0, 2016, 0, -9792, 0, 7936, 1, 10, 0, -240, 0, 4032, 0, -32640, 0, 79360, 0
Offset: 0

Views

Author

Peter Luschny, Jun 04 2023

Keywords

Comments

The Swiss-Knife polynomials (A081658 and A153641) generate the dual triangle ('dual' in the sense of Euler-tangent versus Euler-secant numbers).

Examples

			The triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2, 0;
[3] 1, 3, 0,   -2;
[4] 1, 4, 0,   -8, 0;
[5] 1, 5, 0,  -20, 0,   16;
[6] 1, 6, 0,  -40, 0,   96, 0;
[7] 1, 7, 0,  -70, 0,  336, 0,  -272;
[8] 1, 8, 0, -112, 0,  896, 0, -2176, 0;
[9] 1, 9, 0, -168, 0, 2016, 0, -9792, 0, 7936;

Links

Crossrefs

Cf. A122045 (alternating row sums), A119880 (row sums), A214447 (central column), A155585 (main diagonal), A109573 (subdiagonal), A162660 (variant), A000364.
Cf. A081658, A153641, A220901, A318254, A346838.

Programs

  • Maple
    P := n -> add(binomial(n + 1, j)*bernoulli(j, 1)*(4^j - 2^j)*x^(j-1), j = 0..n+1) / (n + 1):  T := (n, k) -> coeff(P(n), x, k):
seq(seq(T(n, k), k = 0..n), n = 0..9);
# Second program, based on the generating functions of the columns:
ogf := n -> -(-2)^n * euler(n, 1) / (x - 1)^(n + 1):
ser := n -> series(ogf(n), x, 16):
T := (n, k) -> coeff(ser(k), x, n - k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Alternative, based on the bivariate generating function:
egf :=  exp(x*y) * (1 + tanh(y)): ord := 20:
sery := series(egf, y, ord): polx := n -> coeff(sery, y, n):
coefx := n -> seq(n! * coeff(polx(n), x, n - k), k = 0..n):
for n from 0 to 9 do coefx(n) od;
  • Python
    from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k == 0: return 1
    if k % 2 == 0:  return 0
    if k == n: return 1 - sum(T(n, j) for j in range(1, n, 2))
    return (T(n - 1, k) * n) // (n - k)
for n in range(10): print([T(n, k) for k in range(n + 1)])
  • SageMath
    def B(n: int):
    return bernoulli_polynomial(1, n)
def P(n: int):
    return sum(binomial(n + 1, j) * B(j) * (4^j - 2^j) * x^(j - 1)
           for j in range(n + 2)) / (n + 1)
for n in range(10): print(P(n).list())

Formula

For a recursion see the Python program.
T(n, k) = [x^k] P(n, x) where P(n, x) = (1 / (n + 1)) * Sum_{j=0..n+1} binomial(n + 1, j) * Bernoulli(j, 1) * (4^j - 2^j) * x^(j - 1).
Integral_{x=-n..n} P(n, x)/2 dx = n.
T(n, k) = [x^(n - k)] -(-2)^k * Euler(k, 1) / (x - 1)^(k + 1).
T(n, k) = n! * [x^(n - k)][y^n] exp(x*y) * (1 + tanh(y)).

Extensions

Simpler name by Peter Luschny, Nov 17 2024

A318255 Associated Omega numbers of order 3, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 10, -9, 1, 28, -504, 477, 1, 55, -4158, 78705, -74601, 1, 91, -18018, 1432431, -27154764, 25740261, 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817, 1, 190, -139536, 60087690, -12529983960, 997692516360, -18914487631380, 17929265150637
Offset: 0

Views

Author

Peter Luschny, Aug 26 2018

Keywords

Comments

See the comments in A318254.

Examples

			Triangle starts:
[0] 1
[1] 1,   1
[2] 1,  10,     -9
[3] 1,  28,   -504,      477
[4] 1,  55,  -4158,    78705,     -74601
[5] 1,  91, -18018,  1432431,  -27154764,    25740261
[6] 1, 136, -55692, 11595870, -923261976, 17503377480, -16591655817

Crossrefs

T(n, 0) = A060544, T(n, n) = A293951(n+1) (up to signs), row sums are A040000.
Cf. A318146, A318253, A318254 (m=2).

Programs

  • Maple
    # The function TNum is defined in A318253.
T := (m, n, k) -> `if`(k=0, 1, binomial(m*n-1, m*(n-k))*TNum(m, k)):
for n from 0 to 6 do seq(T(3, n, k), k=0..n) od;
  • Sage
    # uses[AssociatedOmegaNumberTriangle from A318254]
A318255Triangle = lambda dim: AssociatedOmegaNumberTriangle(3, dim)
print(A318255Triangle(8))

Formula

T(m, n, k) = binomial(m*n-1, m*(n-k))*A318253(m, k) for k>0 and 1 for k=0. We consider here the case m=3.
Showing 1-3 of 3 results.

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