A027451 -id:A027451 - cp's OEIS frontend

A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

4, 36, 9, 576, 144, 64, 14400, 3600, 1600, 900, 518400, 129600, 57600, 32400, 20736, 25401600, 6350400, 2822400, 1587600, 1016064, 705600, 1625702400, 406425600, 180633600, 101606400, 65028096, 45158400, 33177600, 131681894400

Offset: 1

Johannes W. Meijer, Jul 21 2009

The hypergeometric function 3F2([1,n+1,n+1],[n+2,n+2],z) = (n+1)^2*Li2(z)/z^(n+1) - MN(z;n)/(n!^2*z^n) for n >= 1, with Li2(z) the dilogarithm. The polynomial coefficients of MN(z;n) lead to the triangle given above.
We observe that 3F2([1,1,1],[2,2],z) = Li2(z)/z and that 3F2([1,0,0],[1,1],z) = 1.
The generating function for the EG1[3,n] coefficients of the EG1 matrix, see A162005, is GFEG1(z;m=2) = 1/(1-z)*(3*zeta(3)/2-2*z*log(2)* 3F2([1,1,1],[2,2],z) + sum((2^(1-2*n)* factorial(2*n-1)*z^(n+1)*3F2([1,n+1,n+1],[n+2,n+2],z))/(factorial(n+1)^2), n=1..infinity)).
The zeros of the MN(z;n) polynomials for larger values of n get ever closer to the unit circle and resemble the full moon, hence we propose to call the MN(z;n) the moon polynomials.

			The first few rows of the triangle are:
  [4]
  [36, 9]
  [576, 144, 64]
  [14400, 3600, 1600, 900]
The first few MN(z;n) polynomials are:
  MN(z;n=1) = 4
  MN(z;n=2) = 36 + 9*z
  MN(z;n=3) = 576 + 144*z + 64*z^2
  MN(z;n=4) = 14400 + 3600*z + 1600*z^2 + 900*z^3

Lewin, L., Polylogarithms and Associated Functions. New York, North-Holland, 1981.

Paolo Xausa, Table of n, a(n) for n = 1..5050 (rows 1..100 of the triangle, flattened).
Eric Weisstein's World of Mathematics, Dilogarithm.

A162995 is a scaled version of this triangle.
A001819(n)*(n+1)^2 equals the row sums for n>=1.
A162991 and A162992 equal the first and second right hand columns.
A001048, A052747, A052759, A052778, A052794 are related to the square root of the first five right hand columns.
A001044, A162993 and A162994 equal the first, second and third left hand columns.
A000142, A001710, A002301, A133799, A129923, A001715 are related to the square root of the first six left hand columns.
A027451(n+1) equals the denominators of M(z, n)/(n!)^2.
A129202(n)/A129203(n) = (n+1)^2*Li2(z=1)/(Pi^2) = (n+1)^2/6.
Cf. A002378 and A035287.

a := proc(n, m): ((n+1)!/m)^2 end: seq(seq(a(n, m), m=1..n), n=1..7); # Johannes W. Meijer, revised Nov 29 2012

Table[((n+1)!/m)^2, {n, 10}, {m, n}] (* Paolo Xausa, Mar 30 2024 *)

a(n,m) = ((n+1)!/m)^2 for n >= 1 and 1 <= m <= n.

A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.

Original entry on oeis.org

1, 4, -3, 36, -27, -5, 144, -108, -20, -7, 3600, -2700, -500, -175, -81, 3600, -2700, -500, -175, -81, -44, 176400, -132300, -24500, -8575, -3969, -2156, -1300, 705600, -529200, -98000, -34300, -15876, -8624, -5200, -3375, 6350400, -4762800, -882000, -308700, -142884, -77616, -46800, -30375, -20825

Offset: 1

Wolfdieter Lang, Jul 20 2006

The row polynomials P(n,x) = Sum_{k=1..n-1} a(n,k)*x^k, n >= 1, appear in the numerator of the o.g.f. for column n of the triangle of rationals A120072(m,n)/A120073(m,n), m >= 2, n = 1..m-1. P(n,x) has degree n-1.

See the W. Lang link under A120072 for the precise form of the o.g.f.s: G(x,n) = -dilog(1-x) + x*P(n,4)/*(A(n)*(n^2)*(1-x)), with A(n) = [1, 1, 4, 9, 144, 100, 3600, 11025, 78400, 63504, ...] = conjectured to be A027451(n), n >= 1.

			For n=2 the o.g.f. of A120072(m,2)/A120073(m,2) (=[5/36, 3/16, 21/100, 2/9, ...]) is G(x,2) = -dilog(1-x) + x*P(2,x)/(1*4*(1-x)) = -dilog(1-x) + x*(4-3*x)/(4*(1-x)).
Triangle begins:
       1;
       4,      -3;
      36,     -27,     -5;
     144,    -108,    -20,    -7;
    3600,   -2700,   -500,  -175,   -81;
    3600,   -2700,   -500,  -175,   -81,   -44;
  176400, -132300, -24500, -8575, -3969, -2156, -1300;

G. C. Greubel, Rows n = 1..50 of the triangle, flattened
Wolfdieter Lang, First ten rows

Row sums (unsigned) give A120079.
Signed row sums conjectured to coincide with A027451.
Cf. A027451, A051418, A120072, A120073, A120079.

f:= func< n | n eq 1 select 1 else 1/n^2 -1/(n-1)^2 >;
A120078:= func< n,k | (Lcm([1..n]))^2*f(k) >;
[A120078(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 26 2023

Table[(Apply[LCM, Range[n]])^2*If[k==1, 1, (1-2*k)/(k*(k-1))^2], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 26 2023 *)

def f(k): return 1 if (k==1) else 1/k^2 - 1/(k-1)^2
def A120078(n,k): return (lcm(range(1, n+1)))^2*f(k)
flatten([[A120078(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Apr 26 2023

T(n, k) = A051418(n) * (1 if k = 1 otherwise 1/k^2 - 1/(k-1)^2). - G. C. Greubel, Apr 26 2023

A120079 Unsigned row sums of triangle A120078.

Original entry on oeis.org

1, 7, 68, 279, 7056, 7100, 349200, 1400175, 12622400, 12637296, 1530446400, 1531460700, 258950260800, 259056111600, 259141506624, 1036845584775, 299715332716800, 299771444772800, 108234634597689600, 108249271042728816, 108261866776377600, 108272784263716800

Offset: 1

Wolfdieter Lang, Jul 20 2006

G. C. Greubel, Table of n, a(n) for n = 1..1000

Signed row sums conjectured to be A027451(n), which also appears in the denominator of o.g.f.s. G(x, n) given in A120078 as numbers A(n).

Cf. A000290, A051418, A056220.

[(2-1/n^2)*(Lcm([1..n]))^2: n in [1..40]]; // G. C. Greubel, Apr 26 2023

Table[(2-1/n^2)*(Apply[LCM, Range[n]])^2, {n, 40}] (* G. C. Greubel, Apr 26 2023 *)

def A120079(n): return (2 - 1/n^2)*(lcm(range(1, n+1)))^2
[A120079(n) for n in range(1,41)] # G. C. Greubel, Apr 26 2023

a(n) = Sum_{k=1..n} abs(A120078(n,k)), n >= 1.
From G. C. Greubel, Apr 26 2023: (Start)
a(n) = (2 - 1/n^2)*A051418(n).
a(n) = A056220(n)*A051418(n)/A000290(n). (End)

Terms a(11) onward added by G. C. Greubel, Apr 26 2023

A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

Original entry on oeis.org

1, 8, 108, 576, 18000, 21600, 1234800, 5644800, 57153600, 63504000, 8452382400, 9220780800, 1688171284800, 1818030614400, 1947889944000, 8310997094400, 2551995545299200, 2702112930316800, 1029655143835718400

Offset: 1

Wolfdieter Lang, Jul 20 2006

The triangle of rationals is the matrix cube of the matrix with elements a(i,j) = 1/i if j <= i, 0 if j > i.

Stephen Crowley, Two New Zeta Constants: Fractal String, Continued Fraction, and Hypergeometric Aspects of the Riemann Zeta Function, arXiv:1207.1126 [math.NT], 2012.

Distinct from A246498.

A027447(i,j)= a(i)* A119935(i,j)/A119932(i,j) .
a(n) = lcm_{m=1..n} seq(A119932(n,m)), n >= 1.
a(n)/n^3 = A027451(n) = A002944(n)^2 (the second equation is a conjecture).
a(n)/n^3 = (A099946(n)*(n-1))^2, n >= 2 (from the conjecture).

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A162990 Triangle of polynomial coefficients related to 3F2([1,n+1,n+1],[n+2,n+2],z).

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A120078 Coefficient triangle of numerator polynomials appearing in certain column o.g.f.s related to the H-atom spectrum.

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A120079 Unsigned row sums of triangle A120078.

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A119936 Least common multiple (LCM) of denominators of the rows of the triangle of rationals A119935/A119932.

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