A318146 -id:A318146 - cp's OEIS frontend

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.

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

Peter Luschny, Aug 22 2018

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.

			[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]

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

# 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;

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 *)

# 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)])

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

Peter Luschny, Aug 22 2018

sign

Cf. A318253 (case n=5), A318146.

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

# The function OmegaPolynomial is defined in A318146.
seq(coeff(OmegaPolynomial(5, n), x, 1), n=0..11);

LMlist[m_, len_] := Table[(m n)!, {n, 0, len}]*
CoefficientList[Series[Log[MittagLefflerE[m, z]], {z, 0, len}], z];
LMlist[5, 13]

A088874 T(n, k) = [x^k] (2n)! [z^(2n)] 1/cos(z)^x, triangle read by rows, for 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, 0, 1903757312

Offset: 0

Philippe Deléham, Nov 26 2003

Previous name was: Triangle read by rows, given by [0, 2, 6, 12, 20, 30, 42, 56, ...] DELTA [1, 2, 3, 4, 5, 6, 7, 8, ...] where Delta is the operator defined in A084938.

			Triangle starts:
[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

Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) #09.7.6.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles , JIS 9 (2006) #06.4.1.
Alan D. Sokal, The Euler and Springer numbers as moment sequences, arXiv:1804.04498 [math.CO], 2018.

Another version of the triangle A085734. A signed version is A318146.

Diagonals give: A000007 A000182 A001147, row sums A000364.

ser := series(sec(z)^x, z, 24): row := n -> n!*coeff(ser, z, n):
seq(seq(coeff(row(2*n), x, k), k=0..n), n=0..8); # Peter Luschny, Jul 01 2019

T[1, 1] = 1; T[n_, k_] := Sum[(1/2^(j-1))*StirlingS1[j, k-1]*Sum[(-1)^(i + k + n)*(i-j)^(2(n-1)) Binomial[2j, i], {i, 0, j-1}]/j!, {j, 1, n-1}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 14 2018, after Vladimir Kruchinin *)
a[n_] := (2n)! SeriesCoefficient[Sec[z]^x, {z, 0, 2n}] // CoefficientList[#, x] &;
Table[a[n], {n, 0, 8}] // Flatten (* Peter Luschny, Jul 01 2019 *)

# uses [A241171]
def fr2_row(n):
    if n == 0: return [1]
    S = sum(A241171(n, k)*(x-1)^(n-k) for k in (0..n))
    L = expand(S).list()
    return sum(L[k]*binomial(x+k, n) for k in (0..n-1)).list()
A088874_row = lambda n: [(-1)^(n-k)*m for k,m in enumerate(fr2_row(n))]
for n in (0..7): print(A088874_row(n)) # Peter Luschny, Sep 19 2017

T(n, k) = A085734(n-1, k-1) for n>0 and k>0.

T(n, k) = [x^k] (2*n)! [z^(2*n)] sec(z)^x. - Peter Luschny, Jul 01 2019

New name by Peter Luschny, Jul 01 2019

A318254 Associated Omega numbers of order 2, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, -2, 1, 5, -20, 16, 1, 7, -70, 336, -272, 1, 9, -168, 2016, -9792, 7936, 1, 11, -330, 7392, -89760, 436480, -353792, 1, 13, -572, 20592, -466752, 5674240, -27595776, 22368256, 1, 15, -910, 48048, -1750320, 39719680, -482926080, 2348666880, -1903757312

Offset: 0

Peter Luschny, Aug 26 2018

The Omega polynomials A318146 are defined 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. 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.

			Triangle starts:
[0] [1]
[1] [1,  1]
[2] [1,  3,   -2]
[3] [1,  5,  -20,    16]
[4] [1,  7,  -70,   336,    -272]
[5] [1,  9, -168,  2016,   -9792,    7936]
[6] [1, 11, -330,  7392,  -89760,  436480,   -353792]
[7] [1, 13, -572, 20592, -466752, 5674240, -27595776, 22368256]

Even-indexed rows of A220901 (up to signs).
T(n, 0) = A005408, T(n, n) = A220901 (up to signs), row sums are A040000.
Cf. A318146, A318253, A318255 (m=3).

# 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(2, n, k), k=0..n) od;

def AssociatedOmegaNumberTriangle(m, len):
    R = ZZ[x]; B = [1]*len; L = [R(1)]*len; T = [[1]]
    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))
        T.append([1] + [binomial(m*k-1, m*(k-j))*B[j] for j in (1..k)])
    return T
A318254Triangle = lambda dim: AssociatedOmegaNumberTriangle(2, dim)
print(A318254Triangle(8))

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=2.

A318147 Coefficients of the Omega polynomials of order 3, triangle T(n,k) read by rows with 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, -9, 10, 0, 477, -756, 280, 0, -74601, 142362, -83160, 15400, 0, 25740261, -55429920, 40900860, -12612600, 1401400, 0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400, 190590400

Offset: 0

Peter Luschny, Aug 22 2018

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

			[0] [1]
[1] [0,            1]
[2] [0,           -9,          10]
[3] [0,          477,        -756,          280]
[4] [0,       -74601,      142362,       -83160,       15400]
[5] [0,     25740261,   -55429920,     40900860,   -12612600,     1401400]
[6] [0, -16591655817, 38999319642, -33465991104, 13440707280, -2572970400,190590400]

All row sums are 1, alternating row sums (taken absolute) are A002115.
T(n,1) ~ A293951(n), T(n,n) = A025035(n).
A023531 (m=1), A318146 (m=2), this seq (m=3), A318148 (m=4).

# See A318146 for the missing functions.
FL([seq(CL(OmegaPolynomial(3, n)), n=0..8)]);

(* OmegaPolynomials are defined in A318146 *)
Table[CoefficientList[OmegaPolynomial[3, n], x], {n, 0, 6}] // Flatten

# See A318146 for the function OmegaPolynomial.
[list(OmegaPolynomial(3, n)) for n in (0..6)]

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 = 3 (for other cases see the cross-references).

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

Peter Luschny, Aug 22 2018

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

			[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]

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).

# See A318146 for the missing functions.
FL([seq(CL(OmegaPolynomial(4, n)), n=0..8)]);

(* OmegaPolynomials are defined in A318146 *)
Table[CoefficientList[OmegaPolynomial[4, n], x], {n, 0, 6}] // Flatten

# See A318146 for the function OmegaPolynomial.
[list(OmegaPolynomial(4, n)) for n in (0..6)]

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).

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

Peter Luschny, Aug 26 2018

See the comments in A318254.

			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

T(n, 0) = A060544, T(n, n) = A293951(n+1) (up to signs), row sums are A040000.

Cf. A318146, A318253, A318254 (m=2).

# 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;

# uses[AssociatedOmegaNumberTriangle from A318254]
A318255Triangle = lambda dim: AssociatedOmegaNumberTriangle(3, dim)
print(A318255Triangle(8))

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.

cp's OEIS Frontend

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.

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

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A088874 T(n, k) = [x^k] (2*n)! [z^(2*n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.

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A318254 Associated Omega numbers of order 2, triangle T(n,k) read by rows for n >= 0 and 0 <= k <= n.

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A318147 Coefficients of the Omega polynomials of order 3, triangle T(n,k) read by rows with 0<=k<=n.

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

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

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A088874 T(n, k) = [x^k] (2n)! [z^(2n)] 1/cos(z)^x, triangle read by rows, for 0 <= k <= n.