A002445 -id:A002445 - cp's OEIS frontend

A120081 Denominators of expansion for original Debye function (n=3).

1, 8, 20, 1, 1680, 1, 90720, 1, 4435200, 1, 207567360, 1, 6538371840000, 1, 423437414400, 1, 67580611338240000, 1, 35763659520196608000, 1, 6155242080686899200000, 1, 117509166994931712000000, 1, 15244417230585693025075200000, 1, 1799300365026394374144000000

Offset: 0

Wolfdieter Lang, Jul 20 2006

Numerators are given in A120080.

See A120070 for the definition of the Debye function D(x)=D(3,x) and references and links.

G. C. Greubel, Table of n, a(n) for n = 0..500

Cf. A000367, A002445, A120080.

[Denominator(3*Bernoulli(n)/((n+3)*Factorial(n))): n in [0..50]]; // G. C. Greubel, May 01 2023

max = 26; Denominator[CoefficientList[Integrate[Normal[Series[(3*(t^3/(Exp[t] -1)))/x^3, {t, 0, max}]], {t, 0, x}], x]] (* Jean-François Alcover, Oct 04 2011 *)
Table[Denominator[3*BernoulliB[n]/((n+3)*n!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

def A120081(n): return denominator(3*bernoulli(n)/((n+3)*factorial(n)))
[A120081(n) for n in range(51)] # G. C. Greubel, May 01 2023

a(n) = denominator(r(n)), with r(n) = [x^n]( 1 - 3*x/8 + Sum_{k >= 1} (3*B(2*k)/((2*k+3)*(2*k)!))*x^(2*k) ), where B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator( 3*Bernoulli(n)/((n+3)*n!) ), n >= 0. - G. C. Greubel, May 01 2023

A120086 Numerators of expansion of Debye function for n=4: D(4,x).

Original entry on oeis.org

1, -2, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are found under A120087.

See the W. Lang link under A120080 for more details on the general case D(n,x), n= 1, 2, ... D(4,x) is the e.g.f. of the rational sequence {4*B(n)/(n+4)}, n >= 0. See A227573/A227574. - Wolfdieter Lang, Jul 17 2013

			Rationals r(n): [1, -2/5, 1/18, 0, -1/1440, 0, 1/75600, 0, -1/3628800, 0, 1/167650560, 0, -691/5230697472000, ...].

Vincenzo Librandi, Table of n, a(n) for n = 0..600
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=1, with a factor (x^4)/4 extracted.
Wolfdieter Lang, Rationals r(n).

Cf. A060054. [From R. J. Mathar, Aug 07 2008]
Cf. A000367/A002445, A027641/A027642, A120097, A227573/A227574 (D(4,x) as e.g.f.). - Wolfdieter Lang, Jul 17 2013
Cf. A120080, A120081, A120082, A120083, A120084, A120085, A120087 (denominators).

[Numerator(4*(n+1)*(n+2)*(n+3)*Bernoulli(n)/Factorial(n+4)): n in [0..50]]; // G. C. Greubel, May 02 2023

r[n_]:= 4*BernoulliB[n]/((n+4)*n!); Table[r[n]//Numerator, {n,0,36}] (* Jean-François Alcover, Jun 21 2013 *)

[numerator(4*(n+1)*(n+2)*(n+3)*bernoulli(n)/factorial(n+4)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = numerator(r(n)), with r(n) = [x^n](1 - 4*x/(2*(4+1)) + 2*Sum_{k >= 0} (B(2*k)/((k+2)*(2*k)!))*x^(2*k) ), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = numerator(4*B(n)/((n+4)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). From D(4,x) read as o.g.f. - Wolfdieter Lang, Jul 17 2013

A164020 Denominators of Bernoulli numbers interleaved with even numbers.

Original entry on oeis.org

1, 2, 6, 4, 30, 6, 42, 8, 30, 10, 66, 12, 2730, 14, 6, 16, 510, 18, 798, 20, 330, 22, 138, 24, 2730, 26, 6, 28, 870, 30, 14322, 32, 510, 34, 6, 36, 1919190, 38, 6, 40, 13530, 42, 1806, 44, 690, 46, 282, 48, 46410, 50, 66, 52, 1590, 54, 798, 56, 870, 58, 354, 60, 56786730

Offset: 0

Paul Curtz, Aug 08 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Bernoulli Number

Cf. A027642, A141056, A160014.

[IsEven(n) select Denominator(Bernoulli(n)) else n+1: n in [0..100]]; // Vincenzo Librandi, Sep 08 2017

a[n_]:=If[OddQ[n], n+1, BernoulliB[n] // Denominator]; Table[a[n], {n, 0, 60}](* Jean-François Alcover, Dec 29 2012 *)
With[{nn=60},Riffle[Denominator[BernoulliB[Range[0,nn,2]]],Range[2,nn,2]]] (* Harvey P. Dale, Jul 18 2015 *)

a(2*n) = A002445(n).
a(2*n+1) = 2*(n+1).
a(n) divides A057643(n). Franklin T. Adams-Watters, Aug 03 2012

Extended by R. J. Mathar, Sep 23 2009

A165734 Period 2: repeat 6,30.

Original entry on oeis.org

6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30, 6, 30

Offset: 1

Paul Curtz, Sep 25 2009

Continued fraction expansion of (15+sqrt(230))/5. - Klaus Brockhaus, Apr 28 2010

Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A124886, A002445, A040214.

PadRight[{},120,{6,30}] (* Harvey P. Dale, Jun 03 2019 *)

a(n)=30-n%2*24 \\ Charles R Greathouse IV, Jun 02 2011

From Philippe Deléham, Sep 27 2009: (Start)
G.f.: 6x(1+5x)/((1-x)(1+x)).
a(n) = 6*A010686(n-1). (End)

Edited by N. J. A. Sloane, Sep 25 2009

A369134 Triangle read by rows: T(n, k) = (-1)^(n + 1)L(n) M(n, k) where M is the inverse of the matrix generated by the triangle A368846 and L(n) is the lcm of the denominators of the terms in the n-th row of M.

Original entry on oeis.org

-1, 0, 1, 0, 0, -1, 0, 0, 7, 3, 0, 0, -14, -6, -1, 0, 0, 693, 297, 55, 5, 0, 0, -30030, -12870, -2431, -260, -15, 0, 0, 4150146, 1778634, 337480, 37310, 2625, 105, 0, 0, -21441420, -9189180, -1745458, -194480, -14280, -714, -21

Offset: 0

Peter Luschny, Jan 14 2024

As has been observed by T. Curtright, the absolute value of the nonzero terms in row n of the triangle is monotonically decreasing, and the absolute value of each nonzero term T(n, k) is greater than the sum of the absolute value of the terms in the tail of that row.

The sum of the n-th row divided by the lcm of the n-th row of A368848 is the Bernoulli number B(2*n).

			[0] [-1]
[1] [0, 1]
[2] [0, 0,        -1]
[3] [0, 0,         7,        3]
[4] [0, 0,       -14,       -6,       -1]
[5] [0, 0,       693,      297,       55,       5]
[6] [0, 0,    -30030,   -12870,    -2431,    -260,    -15]
[7] [0, 0,   4150146,  1778634,   337480,   37310,   2625,  105]
[8] [0, 0, -21441420, -9189180, -1745458, -194480, -14280, -714, -21]
.
For n = 5:
(0 + 0 + 693 + 297 + 55 + 5) / 13860 = 5 / 66 = Bernoulli(10).

Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024.

Cf. A368846, A368848, A369135, A369120 (row sums), A369121 (T(n,n)), A369122 (T(n,2)), A000367/A002445 (Bernoulli(2n)).

A368846[n_, k_] := If[k == 0, Boole[n == 0], (-1)^(n + k) 2 Binomial[2 k - 1, n] Binomial[2 n + 1, 2 k]];
Map[# LCM @@ Denominator[#]&, MapIndexed[(-1)^First[#2] Take[#, First[#2]]&, Inverse[PadRight[Table[A368846[n, k], {n, 0, 10}, {k, 0, n}]]]]] (* Paolo Xausa, Jan 15 2024 *)

M = matrix(ZZ, 32, 32, A368846).inverse()
def T(n, k):
    L = (-1)**(n + 1)*lcm(M[n][k].denominator() for k in range(n + 1))
    return L * M[n][k]
for n in range(9):
    print([T(n, k) for k in range(n + 1)])

(Sum_{k=0..n} T(n, k)) / A369135(n) = Bernoulli(2*n).

T(n, 2) / T(n, 3) = 7 / 3 for n >= 3.

A002443 Numerator in Feinler's formula for unsigned Bernoulli number |B_{2n}|.

Original entry on oeis.org

1, 1, 1, 2, 3, 10, 1382, 420, 10851, 438670, 7333662, 51270780, 7090922730, 2155381956, 94997844116, 68926730208040, 1780853160521127, 541314450257070, 52630543106106954746, 15997766769574912140, 10965474176850863126142, 1003264444985926729776060, 35069919669919290536128980

Offset: 0

N. J. A. Sloane

A002443/A002444 = |B_{2n}| (see also A000367/A002445).

a(n) is a nontrivial multiple of A000367(n) if gcd(a(n),A002444(n)) > 1. Furthermore, all terms here are positive, whereas the terms of A000367 retain the sign of B_{2n}, e.g., a(8)/A002444(8) = 10851/1530 is the absolute value of A000367(8)/A002445(8) = -3617/510 = B_{16}. - M. F. Hasler, Jan 05 2016

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 208.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. T. Davis, Tables of the Mathematical Functions, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.]
Index entries for sequences related to Bernoulli numbers.

Cf. A002444, A000367, A002445, A266742, A266743, A266911.

See Davis, Vol. 2, p. 206, second displayed equation, where a(n) appears as c_{2k}. Note that the recurrence for c_{2k} involves an extra term c_1 = 1 (which is not a term of the present sequence), and also the numbers M_i^{2k} given in A266743. However, given that contemporary Computer Algebra Systems can easily calculate Bernoulli numbers, and A002444 has a simple formula, the best way to compute a(n) today is via a(n) = A002444(n)*|B_{2n}|. - N. J. A. Sloane, Jan 08 2016

Name amended following a suggestion from T. D. Noe. - M. F. Hasler, Jan 05 2016

Edited with new definition, further terms, and scan of source by N. J. A. Sloane, Jan 08 2016

A120087 Denominators of expansion of Debye function for n=4: D(4,x).

Original entry on oeis.org

1, 5, 18, 1, 1440, 1, 75600, 1, 3628800, 1, 167650560, 1, 5230697472000, 1, 336259123200, 1, 53353114214400000, 1, 28100018194440192000, 1, 4817145976189747200000, 1, 91657150256046735360000, 1, 11856768957122205686169600000, 1, 1396008903899788738560000000, 1

Offset: 0

Wolfdieter Lang, Jul 20 2006

From Wolfdieter Lang, Jul 17 2013: (Start)
The numerators are given in A120086.
See the link under A120080 for D(n,4) as e.g.f. of 4*B(n)/(n+4) = A227573(n)/A227574(n), n>= 0. (End)

			Rationals r(n): [1, -2/5, 1/18, 0, -1/1440, 0, 1/75600, 0, -1/3628800, 0, 1/167650560, 0, -691/5230697472000, ...].

G. C. Greubel, Table of n, a(n) for n = 0..440

Cf. A000367, A002445, A027641, A027642, A120080, A120081, A120082, A120083, A120084, A120085, A120086, A227573, A227574.

[Denominator(4*(n+1)*(n+2)*(n+3)*Bernoulli(n)/Factorial(n+4)): n in [0..50]]; // G. C. Greubel, May 02 2023

Table[Denominator[4*BernoulliB[n]/((n+4)*n!)], {n,0,50}] (* G. C. Greubel, May 02 2023 *)

[denominator(4*(n+1)*(n+2)*(n+3)*bernoulli(n)/factorial(n+4)) for n in range(51)] # G. C. Greubel, May 02 2023

a(n) = denominator(r(n)), with r(n) = [x^n](1 - 2*x/5 + 2*Sum_{k >= 0}(B(2*k)/((k+2)*(2*k)!))*x^(2*k) ), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).

a(n) = denominator(4*B(n)/((n+4)*n!)), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n). From D(4,x) read as o.g.f. - Wolfdieter Lang, Jul 17 2013

A177427 Numerators of the Inverse Akiyama-Tanigawa transform of the aerated even-indexed Bernoulli numbers 1, 0, 1/6, 0, -1/30, 0, 1/42, ...

Original entry on oeis.org

1, 1, 13, 7, 149, 157, 383, 199, 7409, 7633, 86231, 88331, 1173713, 1197473, 1219781, 620401, 42862943, 43503583, 279379879, 283055551, 57313183, 19328341, 449489867, 1362695813, 34409471059, 34738962067, 315510823603, 45467560829, 9307359944587, 9382319148907, 293103346860157, 147643434162641, 594812856101039, 54448301591149

Offset: 0

Paul Curtz, May 07 2010

These are the numerators of the first row of a Table T(n,k) which contains the even-indexed Bernoulli numbers in the first column: T(2n,0) = A000367(n)/A002445(n), T(2n+1,0)=0, and which generates rows with the Akiyama-Tanigawa transform. (Because the first column is given, the algorithm is an inverse Akiyama-Tanigawa transform.)

These are the absolute values of the numerators of the Taylor expansion of sinh(log(x+1))*log(x+1)at x=0. - Gary Detlefs, Aug 31 2011

			The table T(n,k) of fractions generated by the Akiyama-Tanigawa transform, with the column T(n,0) equal to Bernoulli(n) for even n and equal to 0 for odd n, starts in row n=0 as:
1, 1, 13/12, 7/6, 149/120, 157/120, 383/280, 199/140, ...
0, -1/6, -1/4, -3/10, -1/3, -5/14, -3/8, -7/18, -2/5, -9/22, ...
1/6, 1/6, 3/20, 2/15, 5/42, 3/28, 7/72, 4/45, 9/110, 5/66, ...
0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, 3/55, 15/286, ...
-1/30, -1/30, -3/140, -1/105, 0, 1/140, 49/3960, 8/495, ...
0, -1/42, -1/28, -4/105, -1/28, -29/924, -7/264, -28/1287, -87/5005, ...
1/42, 1/42, 1/140, -1/105, -5/231, -9/308, -343/10296, -1576/45045, ...

L. A. Medina, V. H. Moll, E. S. Rowland, Iterated primitives of logarithmic powers, arXiv:0911.1325, arXiv:0911.1325 [math.NT], 2009-2010.
D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05.

Cf. A177690 (denominators).

t[n_, 0] := BernoulliB[n]; t[1, 0]=0; t[n_, k_] := t[n, k] = (t[n, k-1] + (k-1)*t[n, k-1] - t[n+1, k-1])/k; Table[t[0, k], {k, 0, 33}] // Numerator (* Jean-François Alcover, Aug 09 2012 *)

From Groux Roland, Jan 07 2011: (Start)
T(0,k) = H(k)/2 + 1/(k+1) with H(k) harmonic number of order k.
T(0,k)= -(1/2)*(k+1)*Integral_{x=0..1} x^n*log(x*(1-x)) dx.
G.f.: Sum_{k>=0} T(0,k) x^k = (x-2)*(log(1-x))/(2*x*(1-x)). (End)
T(1,n) = -A191567(n)/A061038(n+2) = -A060819(n)/A145979(n). - Paul Curtz, Jul 19 2011
(T(1,n))^2 = A181318(n)/A061038(n+2). - Paul Curtz, Jul 19 2011, index corrected by R. J. Mathar, Sep 09 2011

A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)2binomial(2k - 1, n) binomial(2n + 1, 2k) for k > 0, and k^n for k = 0.

Original entry on oeis.org

1, 0, 6, 0, 0, 30, 0, 0, -70, 140, 0, 0, 0, -840, 630, 0, 0, 0, 924, -6930, 2772, 0, 0, 0, 0, 18018, -48048, 12012, 0, 0, 0, 0, -12870, 216216, -300300, 51480, 0, 0, 0, 0, 0, -350064, 2042040, -1750320, 218790, 0, 0, 0, 0, 0, 184756, -5542680, 16628040, -9699690, 923780

Offset: 0

Peter Luschny, Jan 07 2024

The row sums of the inverse triangle (A368847/A368848) are the unsigned Bernoulli numbers |B(2n)|. To get the signed Bernoulli numbers B(2n), one only needs to change the sign factor in the definition from (-1)^(n + k) to (-1)^(n + 1).

Conjecture: |Sum_{j=0..k} T(k + j, k)| = A229580(k + 1) for k >= 0.

			[0] [1]
[1] [0, 6]
[2] [0, 0,  30]
[3] [0, 0, -70,  140]
[4] [0, 0,   0, -840,    630]
[5] [0, 0,   0,  924,  -6930,   2772]
[6] [0, 0,   0,    0,  18018,  -48048,   12012]
[7] [0, 0,   0,    0, -12870,  216216, -300300,    51480]
[8] [0, 0,   0,    0,      0, -350064, 2042040, -1750320, 218790]

Paolo Xausa, Table of n, a(n) for n = 0..11475 (rows 0..150 of the triangle, flattened).
Thomas Curtright, Scale Invariant Scattering and the Bernoulli Numbers, arXiv:2401.00586 [math-ph], Jan 2024.
Thomas L. Curtright, Bernoulli Partitions, arXiv:2502.09633 [math.CO], 2025.

Cf. A368847/A368848 (inverse), A369134, A369135, A002457 (main diagonal), A000367/A002445 (Bernoulli(2n)), A229580.

A368846[n_,k_] := If[k==0, Boole[n==0], (-1)^(n+k) 2 Binomial[2k-1, n] Binomial[2n+1, 2k]];
Table[A368846[n, k], {n,0,10}, {k,0,n}] (* Paolo Xausa, Jan 08 2024 *)

def A368846(n, k):
    if k == 0: return k^n
    if k  > n: return 0
    return (-1)^(n + k)*2*binomial(2*k - 1, n)*binomial(2*n + 1, 2*k)
for n in range(10): print([A368846(n, k) for k in range(n+1)])

A002444 Denominator in Feinler's formula for unsigned Bernoulli number |B_{2n}|.

Original entry on oeis.org

1, 6, 30, 84, 90, 132, 5460, 360, 1530, 7980, 13860, 8280, 81900, 1512, 3480, 114576, 117810, 1260, 3838380, 32760, 568260, 1191960, 869400, 236880, 9746100, 525096, 629640, 351120, 198360, 42480, 1362881520, 4324320, 1093950, 33008220, 434700, 843480, 46233287100, 102702600, 1081080

Offset: 0

N. J. A. Sloane

A002443/A002444 = |B_{2n}| (see also A000367/A002445).

H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 208.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. T. Davis, Tables of the Mathematical Functions, Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX. [Annotated scan of pages 204-208 of Volume 2.]
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
Index entries for sequences related to Bernoulli numbers.

Cf. A002443, A000367, A002445, A266742, A266743, A266911.

with(numtheory);
g:=proc(m) local i,n; n:=2*m;
mul(ithprime(i)^floor(n/(ithprime(i)-1)),i=1..pi(n+1));
%/n!;
end;
[seq(g(m),m=0..40)]; # N. J. A. Sloane, Jan 08 2016

a[n_] := Product[Prime[i]^Floor[2n/(Prime[i]-1)], {i, 1, PrimePi[2n+1]}]/(2n)!;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 08 2023 *)

Let p_i denote the i-th prime, and let V(n,i) = floor(n/(prime(i)-1)) = A266742(n,i).
Then a(n) = (Prod_i (p_i)^V(n,i))/n!.
(See Davis, Vol. 2, p. 206, first displayed equation, where a(n) appears as d_{2k}.)

Name amended upon suggestion by T. D. Noe, by M. F. Hasler, Jan 05 2016

Edited with new definition, more terms, and scan of source by N. J. A. Sloane, Jan 08 2016

cp's OEIS Frontend

A120081 Denominators of expansion for original Debye function (n=3).

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A120086 Numerators of expansion of Debye function for n=4: D(4,x).

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A164020 Denominators of Bernoulli numbers interleaved with even numbers.

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A165734 Period 2: repeat 6,30.

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A369134 Triangle read by rows: T(n, k) = (-1)^(n + 1)*L(n) * M(n, k) where M is the inverse of the matrix generated by the triangle A368846 and L(n) is the lcm of the denominators of the terms in the n-th row of M.

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A002443 Numerator in Feinler's formula for unsigned Bernoulli number |B_{2n}|.

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A120087 Denominators of expansion of Debye function for n=4: D(4,x).

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Programs

A369134 Triangle read by rows: T(n, k) = (-1)^(n + 1)L(n) M(n, k) where M is the inverse of the matrix generated by the triangle A368846 and L(n) is the lcm of the denominators of the terms in the n-th row of M.

A368846 Triangle read by rows: T(n, k) = (-1)^(n + k)2binomial(2k - 1, n) binomial(2n + 1, 2k) for k > 0, and k^n for k = 0.