A074599 -id:A074599 - cp's OEIS frontend

A025547 Least common multiple of {1,3,5,...,2n-1}.

1, 3, 15, 105, 315, 3465, 45045, 45045, 765765, 14549535, 14549535, 334639305, 1673196525, 5019589575, 145568097675, 4512611027925, 4512611027925, 4512611027925, 166966608033225, 166966608033225, 6845630929362225, 294362129962575675, 294362129962575675

Offset: 1

Clark Kimberling

This sequence coincides with the sequence f(n) = denominator of 1 + 1/3 + 1/5 + 1/7 + ... + 1/(2n-1) iff n <= 38. But a(39) = 6414924694381721303722858446525, f(39) = 583174972216520118520259858775. - T. D. Noe, Aug 04 2004 [See A350670(n-1).]
Coincides for n=1..42 with the denominators of a series for Pi*sqrt(2)/4 and then starts to differ. See A127676.
a(floor((n+1)/2)) = gcd(a(n), A051426(n)). - Reinhard Zumkeller, Apr 25 2011
A051417(n) = a(n+1)/a(n).

T. D. Noe, Table of n, a(n) for n = 1..200
Yue-Wu Li and Feng Qi, A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments, Axioms (2024) Vol. 13, Art. No. 317. See p. 11 of 24.
Eric Weisstein's World of Mathematics, Jeep Problem, Pi, Pi Continued Fraction, Least Common Multiple
Wikipedia, Least common multiple
Index entries for sequences related to lcm's

Cf. A007509, A025550, A075135. The numerators are in A074599.
Cf. A003418 (LCM of {1..n}).
Cf. A005408, A350670.

a025547 n = a025547_list !! (n-1)
a025547_list = scanl1 lcm a005408_list
-- Reinhard Zumkeller, Oct 25 2013, Apr 25 2011

A025547:=proc(n) local i,t1; t1:=1; for i from 1 to n do t1:=lcm(t1,2*i-1); od: t1; end;
f := n->denom(add(1/(2*k-1),k=0..n)); # a different sequence!

a = 1; Join[{1}, Table[a = LCM[a, n], {n, 3, 125, 2}]] (* Zak Seidov, Jan 18 2011 *)
nn=30;With[{c=Range[1,2*nn,2]},Table[LCM@@Take[c,n],{n,nn}]] (* Harvey P. Dale, Jan 27 2013 *)

a(n)=lcm(vector(n,k,2*k-1)) \\ Charles R Greathouse IV, Nov 20 2012

# generates initial segment of sequence
from math import gcd
from itertools import accumulate
def lcm(a, b): return a * b // gcd(a, b)
def aupton(nn): return list(accumulate((2*i+1 for i in range(nn)), lcm))
print(aupton(23)) # Michael S. Branicky, Mar 28 2022

A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

Original entry on oeis.org

1, 1, 2, 3, 8, 4, 15, 46, 36, 8, 105, 352, 344, 128, 16, 945, 3378, 3800, 1840, 400, 32, 10395, 39048, 48556, 27840, 8080, 1152, 64, 135135, 528414, 709324, 459032, 160720, 31136, 3136, 128

Offset: 0

Johannes W. Meijer, Jun 08 2009, Jul 22 2011

The series expansion of (1-x)^((-1-2*n)/2) = sum(b(p)*x^p, p=0..infinity) for n = 0, 1, 2, .. can be described with b(p) = (F(p,n)/ (2*n-1)!!)*(binomial(2*p,p)/4^(p)) with F(x,n) = 2^n * product( x+(2*k-1)/2, k=1..n). The roots of the F(x,n) polynomials can be found at p = (1-2*k)/2 with k from 1 to n for n = 0, 1, 2, .. . The coefficients of the F(x,n) polynomials lead to the triangle given above. The triangle row sums lead to A001147.
Quite surprisingly we discovered that sum(b(p)*x^p, p=0..infinity) = (1-x)^(-1-2*n)/2, for n = -1, -2, .. . We assume that if m = n+1 then the value returned for product(f(k), k = m..n) is 1 and if m> n+1 then 1/product(f(k), k=n+1..m-1) is the value returned. Furthermore (1-2*n)!! = (-1)^(n+1)/(2*n-3)!! for n = 1, 2, 3 .. . This leads to b(p) = ((-1-2*n)!!/ G(p,n))*(binomial(2*p,p) /4^(p)) for n = -1, -2, .. . For the G(p,n) polynomials we found that G(p,n) = F(-p,-n). The roots of the G(p,n) polynomials can be found at p=(2*k-1)/2 with k from 1 to (-n) for n = -1, -2, .. . The coefficients of the G(p,n) polynomials lead to a second triangle that stands with its head on top of the first one. It is remarkable that the row sums lead once again to A001147.
These two triangles together look like an hourglass so we propose to call the F(p,n) and the G(p,n) polynomials the hourglass polynomials.
Triangle T(n,k), read by rows, given by (1, 2, 3, 4, 5, 6, 7, 8, 9, ...) DELTA (2, 0, 2, 0, 2, 0, 2, 0, 2, ...) where DELTA is the operator defined in A084938. Philippe Deléham, May 14 2015.

			From _Gary W. Adamson_, Jul 19 2011: (Start)
The first few rows of matrix M are:
  1, 2,  0,  0, 0, ...
  1, 3,  2,  0, 0, ...
  1, 4,  5,  2, 0, ...
  1, 5,  9,  7, 2, ...
  1, 6, 14, 16, 9, ... (End)
The first few G(p,n) polynomials are:
  G(p,-3) = 15 - 46*p + 36*p^2 - 8*p^3
  G(p,-2) = 3 - 8*p + 4*p^2
  G(p,-1) = 1 - 2*p
The first few F(p,n) polynomials are:
  F(p,0) = 1
  F(p,1) = 1 + 2*p
  F(p,2) = 3 + 8*p + 4*p^2
  F(p,3) = 15 + 46*p + 36*p^2 + 8*p^3
The first few rows of the upper and lower hourglass triangles are:
  [15, -46, 36, -8]
  [3, -8, 4]
  [1, -2]
  [1]
  [1, 2]
  [3, 8, 4]
  [15, 46, 36, 8]

Cf. A001790 [(1-x)^(-1/2)], A001803 [(1-x)^(-3/2)], A161199 [(1-x)^(-5/2)] and A161201 [(1-x)^(-7/2)].
Cf. A002596 [(1-x)^(1/2)], A161200 [(1-x)^(3/2)] and A161202 [(1-x)^(5/2)].
A046161 gives the denominators of the series expansions of all (1-x)^((-1-2*n)/2).
A028338 is a scaled triangle version, A039757 is a scaled signed triangle version and A109692 is a transposed scaled triangle version.
A001147 is the first left hand column and equals the row sums.
A004041 is the second left hand column divided by 2, A028339 is the third left hand column divided by 4, A028340 is the fourth left hand column divided by 8, A028341 is the fifth left hand column divided by 16.
A000012, A000290, A024196, A024197 and A024198 are the first (n-m=0), second (n-m=1), third (n-m=2), fourth (n-m=3) and fifth (n-m=4) right hand columns divided by 2^m.
A074599 * A025549 is not always equals the second left hand column.
Cf. A029635. [Gary W. Adamson, Jul 19 2011]

nmax:=7; for n from 0 to nmax do a(n,n):=2^n: a(n,0):=doublefactorial(2*n-1) od: for n from 2 to nmax do for m from 1 to n-1 do a(n,m) := 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) od: od: seq(seq(a(n,k), k=0..n), n=0..nmax);
nmax:=7: M := Matrix(1..nmax+1,1..nmax+1): A029635 := proc(n,k): binomial(n,k) + binomial(n-1,k-1) end: for i from 1 to nmax do for j from 1 to i+1 do M[i,j] := A029635(i,j-1) od: od: for n from 0 to nmax do B := M^n: for m from 0 to n do a(n,m):= B[1,m+1] od: od: seq(seq(a(n,m), m=0..n), n=0..nmax);
A161198 := proc(n,k) option remember; if k > n or k < 0 then 0 elif n = 0 and k = 0 then 1 else 2*A161198(n-1, k-1) + (2*n-1)*A161198(n-1, k) fi end:
seq(print(seq(A161198(n,k), k = 0..n)), n = 0..6);  # Peter Luschny, May 09 2013

nmax = 7; a[n_, 0] := (2*n-1)!!; a[n_, n_] := 2^n; a[n_, m_] := a[n, m] = 2*a[n-1, m-1]+(2*n-1)*a[n-1, m]; Table[a[n, m], {n, 0, nmax}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014, after Maple *)

for(n=0,9, print(Vec(Ser( 2^n*prod( k=1,n, x+(2*k-1)/2 ),,n+1))))  \\ M. F. Hasler, Jul 23 2011

@CachedFunction
def A161198(n,k):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return 2*A161198(n-1,k-1)+(2*n-1)*A161198(n-1,k)
for n in (0..6): [A161198(n,k) for k in (0..n)]  # Peter Luschny, May 09 2013

a(n,m) := coeff(2^(n)*product((x+(2*k-1)/2),k=1..n), x, m) for n = 0, 1, .. ; m = 0, 1, .. .
a(n, m) = 2*a(n-1,m-1)+(2*n-1)*a(n-1,m) with a(n, n) = 2^n and a(n, 0) = (2*n-1)!!.
a(n,m) = the (m+1)-th term in the top row of M^n, where M is an infinite square production matrix; M[i,j] = A029635(i,j-1) = binomial(i, j-1) + binomial(i-1, j-2) with A029635 the (1.2)-Pascal triangle, see the examples and second Maple program. [Gary W. Adamson, Jul 19 2011]
T(n,k) = 2^k * A028338(n,k). - Philippe Deléham, May 14 2015

A056053 a(n) = smallest odd number 2m+1 such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

Original entry on oeis.org

1, 3, 15, 113, 837, 6183, 45691, 337607, 2494595, 18432707, 136200301, 1006391657, 7436284415, 54947122715, 406007372211, 3000011249847, 22167251422541, 163795064320249, 1210290918990281, 8942907496445513, 66079645178783351, 488266205223462461, 3607826381608149807

Offset: 0

Robert G. Wilson v, Jul 25 2000 and Jan 11 2004

nonn

a(2) = 15 and a(3) = 113 are related to the Borwein integrals. Concretely, a(2) = 15 is the smallest odd m such that the integral Integral_{x=-oo..oo} Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi, and a(3) = 113 is the smallest odd m such that the integral Integral_{x=-oo..oo} cos(x) * Product_{1<=k<=m, k odd} (sin(k*x)/(k*x)) dx is slightly less than Pi/2. See the Wikipedia link and the 3Blue1Brown video link below. - Jianing Song, Dec 10 2022

Calvin C. Clawson, "Mathematical Mysteries, The Beauty and Magic of Numbers," Plenum Press, NY and London, 1996, page 64.

Robert G. Wilson v, Table of n, a(n) for n = 0..500
Grant Sanderson, Researchers thought this was a bug (Borwein integrals), 3Blue1Brown video (2022).
Wikipedia, Borwein integral

Cf. A002387, A056054, A091463, A091464, A091465.

Cf. also A092318, A092317, A092315, A281355, A074599, A025547.

s = 0; k = 1; Do[ While[s = N[s + 1/k, 24]; s <= n, k += 2]; Print[k]; k += 2, {n, 1, 11}]

a(n) ~ floor((1/2)*A002387(2n)).
The next term is approximately the previous term * e^2.
a(n) = A092315(n)*2 + 1 = floor(exp(n*2-Euler)/4+1/8)*2+1 for all n (conjectured). - M. F. Hasler, Jan 24 2017
a(n) ~ exp(2*n - A350763) = (1/2)*exp(2*n - gamma), gamma = A001620. - A.H.M. Smeets, Apr 15 2022

Corrected by N. J. A. Sloane, Feb 16 2004
More terms from Robert G. Wilson v, Apr 17 2004
a(17) corrected - see correction in A092315. - Gerhard Kirchner, Jul 25 2020
a(0) prepended by Robert G. Wilson v, Oct 23 2024

A092315 a(n) is the smallest m such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

Original entry on oeis.org

1, 7, 56, 418, 3091, 22845, 168803, 1247297, 9216353, 68100150, 503195828, 3718142207, 27473561357, 203003686105, 1500005624923, 11083625711270, 81897532160124, 605145459495140, 4471453748222756, 33039822589391675, 244133102611731230, 1803913190804074903

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

From Gerhard Kirchner, May 21 2020: (Start)
The terms a(n), evaluated by the formula, should pass the test OH(a(n))=n and OH(a(n)-1)=n-1, where OH(m) is the odd harmonic series, see above.
Another formula, see link Asymptotic formulas, formula 1, is OH(m) = (log(4*m)+gamma)/2+1/(2*m)-11/(48*m^2)+1/(8*m^3)-127*t/(1920*m^4), 0The Maxima code includes both tests and creates a b-file in the current directory. For n<=1000, the case "Precision too low" does not occur. (End)
a(2) = 7 and a(3) = 56 are related to the Borwein integrals. Concretely, a(2) = 7 is the smallest m such that the integral Integral_{x=-oo..oo} Product_{k=0..m} (sin((2*k+1)*x)/((2*k+1)*x)) dx is slightly less than Pi, and a(3) = 56 is the smallest m such that the integral Integral_{x=-oo..oo} cos(x) * Product_{k=0..m} (sin((2*k+1)*x)/((2*k+1)*x)) dx is slightly less than Pi/2. See the Wikipedia link and the 3Blue1Brown video link below. - Jianing Song, Dec 10 2022

Gerhard Kirchner, Table of n, a(n) for n = 1..1000
Gerhard Kirchner, Asymptotic formulas
Grant Sanderson, Researchers thought this was a bug (Borwein integrals), 3Blue1Brown video (2022).
Wikipedia, Borwein integral

Except for first term, same as A092318. Equals (A056053-1)/2.

Cf. A074599, A025547, A281355.

A092315[n_] := Floor[Exp[2*n - EulerGamma]/4]; Table[A092315[n], {n, 1, 22}] (* Robert P. P. McKone, Jul 13 2021 *)

block(
fpprec:1000, gam: %gamma, nmax:1000,
fl: openw("bfile1000.txt"),
OH(k,t):=(log(4*k)+gam)/2+1/(2*k)-11/(48*k^2)+1/(8*k^3)-127*t/(1920*k^4),
printf(fl, "1 1"),   newline(fl),
for n from 2 thru nmax do
(u: bfloat(exp(2*n-gam)/4), k: floor(u),
x0: bfloat(OH(k,0)), x01: bfloat(OH(k,1)), x1: bfloat(OH(k-1,0)),
n0: floor(x0), n01: floor(x01), n1: floor(x1),  m: n,
if n0=n and n01=n and n1=n-1 then
         (h: concat(n, " ", k), printf(fl, h),  newline(fl)) else n: nmax),
if mGerhard Kirchner, Jul 23 2020 */
/* The first nmax terms are saved as a b-file */

a(n) = floor(exp(2*n-gamma)/4+1/8) for all n >= 1 (conjectured; see also comments in A002387). - M. F. Hasler, Jan 22 2017

a(n) = floor(exp(2*n-gamma)/4). - Gerhard Kirchner, Jul 23 2020

More terms from M. F. Hasler, Jan 24 2017

a(17) in the data section and 127 terms in the b-file corrected by Gerhard Kirchner, Jul 23 2020

A092318 a(n) = smallest m such that value of odd harmonic series Sum_{j=0..m} 1/(2j+1) is >= n.

Original entry on oeis.org

0, 7, 56, 418, 3091, 22845, 168803, 1247297, 9216353, 68100150, 503195828, 3718142207, 27473561357, 203003686105, 1500005624923, 11083625711270, 81897532160124, 605145459495140, 4471453748222756, 33039822589391675

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (127 terms corrected by Gerhard Kirchner)

Apart from first term, same as A092315. Equals (A092317-1)/2.
Cf. A074599, A025547.
Cf. A281355 (= a(n) + 1) for a variant.

a[n_] := Floor[(Exp[2 n - EulerGamma] + 1/2)/4]; a[1] = 0; Array[a, 20] (* Robert G. Wilson v, Jan 25 2017 *)

A092318=n->floor(exp(2*n-Euler)/4+1/8)-(n<2) \\ Cf. comments in A092315. - M. F. Hasler, Jan 24 2017

a(n) = floor(exp(2*n-gamma)/4+1/8), for all n > 1. - M. F. Hasler and Robert G. Wilson v, Jan 22 2017

a(n) = floor(exp(2*n-gamma)/4), for all n > 1, see correction in A092315, Gerhard Kirchner, Jul 25 2020

More terms (computed from A092317) from M. F. Hasler, Jan 22 2017

a(17) corrected by Gerhard Kirchner, Jul 26 2020

A025549 a(n) = (2n-1)!!/lcm{1,3,5,...,2n-1}.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 3, 45, 45, 45, 945, 945, 4725, 42525, 42525, 42525, 1403325, 49116375, 49116375, 1915538625, 1915538625, 1915538625, 86199238125, 86199238125, 603394666875, 30773128010625, 30773128010625, 1692522040584375, 96473756313309375, 96473756313309375

Offset: 1

Clark Kimberling

Michael De Vlieger, Table of n, a(n) for n = 1..569

Not always equal to the second left hand column of A161198 triangle divided by A074599. - Johannes W. Meijer, Jun 08 2009

Cf. A196274 (run lengths of equal terms).

seq(doublefactorial(2*n-1)/lcm(seq((2*k-1), k=1..n)), n=1..27) ; # Johannes W. Meijer, Jun 08 2009

L[ {x___} ] := LCM[ x ]; Table[ (2n-1)!!/L[ Range[ 1, 2n-1, 2 ] ], {n, 1, 50} ]
(* Second program: *)
Array[#!!/LCM @@ Range[1, #, 2] &[2 # - 1] &, 30] (* Michael De Vlieger, Feb 19 2019 *)

a(n) = (((2*n)!/n!)/2^n)/lcm(vector(n, i, 2*i-1)); \\ Michel Marcus, Dec 02 2014

a(n) = A001147(n)/A025547(n). - Michel Marcus, Dec 02 2014

Description corrected and sequence extended by Erich Friedman

More terms from Michel Marcus, Dec 02 2014

A092317 a(n) = smallest odd number 2m+1 such that the partial sum Sum_{j=0..m} 1/(2j+1) of the odd harmonic series is >= n.

Original entry on oeis.org

1, 15, 113, 837, 6183, 45691, 337607, 2494595, 18432707, 136200301, 1006391657, 7436284415, 54947122715, 406007372211, 3000011249847, 22167251422541, 163795064320249, 1210290918990281, 8942907496445513, 66079645178783351

Offset: 1

N. J. A. Sloane, Feb 16 2004

nonn

Except for first term, same as A056053. Equals 2*A092318 + 1. Cf. A074599, A025547

a(n) ~ C*exp(2n) with C = 0.2807297417834425... - M. F. Hasler, Jan 22 2017

More terms (via A056053) from M. F. Hasler, Jan 22 2017

a(17) corrected - see correction in A092315. Gerhard Kirchner, Jul 25 2020

A260630 Numerators of first derivatives of Catalan numbers (as continuous functions of n).

Original entry on oeis.org

-1, 1, 5, 59, 449, 1417, 16127, 429697, 437705, 7549093, 145103527, 146489197, 3396112211, 2442184933, 7369048679, 429556076057, 13374954901367, 13427048535167, 94315062045929, 3500487562166393, 3510273150915593, 144285489968702713, 6218562602767668259

Offset: 0

Vladimir Reshetnikov, Nov 11 2015

Let C(n) = 4^n*Gamma(n+1/2)/(sqrt(Pi)*Gamma(n+2)), then C'(n) = C(n)*(H(n-1/2) - H(n+1) + log(4)), where H(n) = Sum_{k>=1} (1/k-1/(n+k)) are harmonic numbers.

			For n = 3, C'(3) = 59/12, so a(3) = numerator(59/12) = 59.

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A260631 (denominators).

Cf. A000108, A001008, A074599.

Numerator@FunctionExpand@Table[CatalanNumber'[n] , {n, 0, 22}]

a(n) = numerator(d(n)), where d(n) satisfies recurrence: d(0) = -1, d(1) = 1/2, (n+1)^2*d(n) = 2*(4*n^2-2*n-1)*d(n-1) - 4*(2*n-3)^2*d(n-2).

A370692 Square array read by upward antidiagonals: T(n, k) = numerator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).

Original entry on oeis.org

0, 2, 0, 8, -4, 0, 46, -40, 16, 0, 352, -1036, 448, -96, 0, 1126, -51664, 56432, -2624, 768, 0, 13016, -469876, 19410176, -1642592, 62464, -7680, 0, 176138, -57251896, 524760752, -3945483392, 195262208, -1868800, 92160, 0, 176138, -57251896, 524760752, -3945483392, 195262208, -1868800, 92160

Offset: 0

Thomas Scheuerle, Apr 21 2024

			array begins:
0,      0,        0,             0,                0
2,     -4,        16,           -96,               768
8,     -40,       448,          -2624,             62464
46,    -1036,     56432,        -1642592,          195262208
352,   -51664,    19410176,     -3945483392,       3281966329856
1126,  -469876,   524760752,    -319632174752,     797531263755008
13016, -57251896, 698956654912, -4680049729764032, 128444001508242193408

Cf. A370691 (denominators).
Cf. A074599 (first column), A173945 (second column).
Cf. A255008 (denominators polygamma(n, 1) - polygamma(n, k)).
Cf. A255009 (numerators polygamma(n, 1) - polygamma(n, k)).

A := (n, k) -> Psi(k, n + 1/2) - Psi(k, 1/2):
seq(lprint(seq(numer(A(n, k)), k = 0..4)), n=0..6);  # Peter Luschny, Apr 22 2024

T(n, k) = numerator(sum(m=1, n, 1/(2*m-1)^(k+1))*k!*(-2)^k*2)

T(n, k) = numerator( polygamma(k, n + 1/2) - polygamma(k, 1/2) ).
T(n, k) = numerator( k!*(-1)^(k+1)*(zeta((k+1), 1/2 + n) - zeta((k+1), 1/2)) ), where zeta is the Hurwitz zeta function.
T(n, 0) = A074599(n).
T(n, 1) = A173945(n+1).

Showing 1-9 of 9 results.

cp's OEIS Frontend

A025547 Least common multiple of {1,3,5,...,2n-1}.

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A161198 Triangle of polynomial coefficients related to the series expansions of (1-x)^((-1-2*n)/2).

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A056053 a(n) = smallest odd number 2m+1 such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

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A092315 a(n) is the smallest m such that the partial sum of the odd harmonic series Sum_{j=0..m} 1/(2j+1) is > n.

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A092318 a(n) = smallest m such that value of odd harmonic series Sum_{j=0..m} 1/(2j+1) is >= n.

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A025549 a(n) = (2n-1)!!/lcm{1,3,5,...,2n-1}.

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A092317 a(n) = smallest odd number 2m+1 such that the partial sum Sum_{j=0..m} 1/(2j+1) of the odd harmonic series is >= n.

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Formula

A370692 Square array read by upward antidiagonals: T(n, k) = numerator( 2k!(-2)^kSum_{m=1..n}( 1/(2m-1)^(k+1) ) ).