A293951 -id:A293951 - cp's OEIS frontend

A273352 a(n) = 2^(2n+2) F(n) where F(n) is Ramanujan's F(n) = Sum_{k>=1} k^(4n-1)/(e^(Pik)-1) - 16^n Sum_{k>=1} k^(4n-1)/(e^(4Pik)-1).

1, 34, 11056, 14873104, 56814228736, 495812444583424, 8575634961418940416, 265929039218907754399744, 13722623393637762299131396096, 1112372064432735526930220874072064, 135292015985218004848567636630910795776, 23782283324940089109797537284278352042000384

Offset: 1

Marko Riedel, May 20 2016

nonn

Bisection of the reduced tangent numbers, A002105. This follows from the formulas. - Franklin T. Adams-Watters, May 22 2016

Math.Stackexchange.Com, Marko R. Riedel et al., Closed form of a sum by Ramanujan

Cf. A002105.

Cf. A000182 (m=2), A293951 (m=3), this seq (m=4), A318258 (m=5).

S := proc(n, k) option remember;
if k=0 then `if`(n=0, 1, 0) else S(n, k-1) + S(n-1, n-k) fi end:
A273352 := n -> S(4*n-1, 4*n-1)/2^(2*n-1):
seq(A273352(n), n=1..12); # Peter Luschny, Jan 18 2017

Table[2^(2*n + 2)*BernoulliB[4*n]*(1 - 2^(4*n))/(8*n), {n, 1, 10}] (* G. C. Greubel, May 21 2016 *)
(* Function LMLlist defined in A293951 *)
LMLlist[4, 13] (* Peter Luschny, Aug 26 2018 *)

a(n) = 2^{2*n+2} * Bernoulli(4*n) * (1-2^(4*n))/(8*n).

A293950 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type A_n joins.

Original entry on oeis.org

1, 2, 40, 3194, 666160, 287316122, 222237912664, 280180369563194, 537546603651987424, 1490424231594917313242, 5735930050702709579598280, 29665120602262869704075225594, 200776204583497637546703088929040, 1739317344037357729592558320480494362

Offset: 0

N. J. A. Sloane, Oct 29 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 0..100
Jeffrey Remmel, S. Zheng, Rises in forests of binary shrubs, arXiv preprint arXiv:1611.09018 [math.CO], 2016-2017.

Cf. A293951, A293952, A293953.

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

A293952 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type E_n joins.

Original entry on oeis.org

1, 3, 99, 11259, 3052323, 1620265923, 1488257158851, 2172534146099019, 4736552519729393091, 14708695606607601165843, 62671742039942099631403299, 355493170380387030721038571419, 2618304731622723256262677112102883, 24521387779014982719407393681918617443

Offset: 0

N. J. A. Sloane, Oct 29 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 0..100
Jeffrey Remmel, S. Zheng, Rises in forests of binary shrubs, arXiv preprint arXiv:1611.09018 [math.CO], 2016-2017.

Cf. A293950, A293951, A293953.

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

A293953 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type S_n joins.

Original entry on oeis.org

1, 5, 169, 19241, 5216485, 2769073949, 2543467934449, 3712914075133121, 8094884285992309261, 25137521105896509819605, 107107542395866078895709049, 607546536582495444795794603801, 4474746926116218974619451907008885, 41907652406924609850893820950967654989

Offset: 0

N. J. A. Sloane, Oct 29 2017

nonn

Lars Blomberg, Table of n, a(n) for n = 0..100
Jeffrey Remmel, S. Zheng, Rises in forests of binary shrubs, arXiv preprint arXiv:1611.09018 [math.CO], 2016-2017.

Cf. A293950, A293951, A293952.

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

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]

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).

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

A273352 a(n) = 2^(2n+2) F(n) where F(n) is Ramanujan's F(n) = Sum_{k>=1} k^(4n-1)/(e^(Pi*k)-1) - 16^n* Sum_{k>=1} k^(4n-1)/(e^(4*Pi*k)-1).

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A293950 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type A_n joins.

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Extensions

A293952 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type E_n joins.

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Author

Keywords

Links

Crossrefs

Extensions

A293953 Number of linear extensions of a poset whose Hasse diagram consists of n binary shrubs with type S_n joins.

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Author

Keywords

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Crossrefs

Extensions

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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Maple

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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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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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Maple

Sage

Formula

A273352 a(n) = 2^(2n+2) F(n) where F(n) is Ramanujan's F(n) = Sum_{k>=1} k^(4n-1)/(e^(Pik)-1) - 16^n Sum_{k>=1} k^(4n-1)/(e^(4Pik)-1).