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
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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 *)
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).
Original entry on oeis.org
1, 34, 11056, 14873104, 56814228736, 495812444583424, 8575634961418940416, 265929039218907754399744, 13722623393637762299131396096, 1112372064432735526930220874072064, 135292015985218004848567636630910795776, 23782283324940089109797537284278352042000384
Offset: 1
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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
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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 *)
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
[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]
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# 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;
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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 *)
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# 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)])
Showing 1-3 of 3 results.
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