A129203 -id:A129203 - 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.

A129196 a(n) = denominator(3*(3+(-1)^n)/(n+1)^3).

Original entry on oeis.org

1, 4, 9, 32, 125, 36, 343, 256, 243, 500, 1331, 288, 2197, 1372, 1125, 2048, 4913, 972, 6859, 4000, 3087, 5324, 12167, 2304, 15625, 8788, 6561, 10976, 24389, 4500, 29791, 16384, 11979, 19652, 42875, 7776, 50653, 27436, 19773, 32000, 68921, 12348, 79507, 42592

Offset: 0

Paul Barry, Apr 02 2007, Apr 03 2007

Numerator of 3*(3+(-1)^n)/(n+1)^3 is A129197.

(1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*(n+1)*t)*((t-Pi)/i)^3 dt = (A129202(n)*Pi^2-A129203(n))/A129196(n) with i=sqrt(-1).

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,4,0,0,0,0,0,-6,0,0,0,0,0,4,0,0,0,0,0,-1).

Cf. A129196, A129197, A129202, A129203, A129204.

a[n_] := Denominator[3*(3 + (-1)^n)/(n + 1)^3]; Array[a, 50, 0] (* Amiram Eldar, Sep 11 2022 *)

a(n) = A129204(n+1)/(5/3+(4/3)*cos(2*Pi*(n+1)/3)).
a(n) = denominator((1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*(n+1)*t)*((t-Pi)/i)^3 dt) with i=sqrt(-1).
a(n) = denominator((Pi^2*(n+1)^2-6)/(n+1)^3).
a(n) = ((n+1)^3/(gcd(n+1,2)*gcd(n+1,3))). - Paul Barry, Oct 09 2007
a(n) = numerator of coefficient of x^6 in the Maclaurin expansion of -exp(-(n+1)*x^2). - Francesco Daddi, Aug 04 2011
Sum_{n>=0} 1/a(n) = 29*zeta(3)/24. - Amiram Eldar, Sep 11 2022

A283393 a(n) = gcd(n^2-1, n^2+9).

Original entry on oeis.org

1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10

Offset: 0

Bruno Berselli, Mar 07 2017

Periodic with period 10.
Similar sequences with formula gcd(n^2-1, n^2+k):
k= 1:  1,  2, 1, 2, 1,  2, 1,  2, 1,  2, 1,  2, 1, ...  (A000034)
k= 3:  1,  4, 1, 4, 1,  4, 1,  4, 1,  4, 1,  4, 1, ...  (A010685)
k= 5:  1,  6, 3, 2, 3,  6, 1,  6, 3,  2, 3,  6, 1, ...  (A129203, start 6)
k= 7:  1,  8, 1, 8, 1,  8, 1,  8, 1,  8, 1,  8, 1, ...  (A010689)
k= 9:  1, 10, 1, 2, 5,  2, 5,  2, 1, 10, 1, 10, 1, ...  (this sequence)
k=11:  1, 12, 3, 4, 3, 12, 1, 12, 3,  4, 3, 12, 1, ...  (A129197, start 12)
Connection between the values of a(n) and the last digit of n:
. if n ends with 0, 2 or 8, then a(n) = 1;
. if n ends with 1 or 9, then a(n) = 10;
. if n ends with 3, 5 or 7, then a(n) = 2;
. if n ends with 4 or 6, then a(n) = 5.
Also, continued fraction expansion of (57 + sqrt(4579))/114.

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

Cf. A000034, A010685, A010689, A129197, A129203.

&cat [[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]^^10];

Table[PolynomialGCD[n^2 - 1, n^2 + 9], {n, 0, 100}]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 10, 1, 2, 5, 2, 5, 2, 1, 10}, 100]

makelist(gcd(n^2-1, n^2+9), n, 0, 100);

Vec((1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10) + O(x^100)) \\ Colin Barker, Mar 08 2017

Python
```
[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]*10
```

[gcd(n^2-1, n^2+9) for n in range(100)]

G.f.: (1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10).

A129202 Denominator of 3*(3+(-1)^n) / (n+1)^2.

Original entry on oeis.org

1, 2, 3, 8, 25, 6, 49, 32, 27, 50, 121, 24, 169, 98, 75, 128, 289, 54, 361, 200, 147, 242, 529, 96, 625, 338, 243, 392, 841, 150, 961, 512, 363, 578, 1225, 216, 1369, 722, 507, 800, 1681, 294, 1849, 968, 675, 1058, 2209, 384, 2401, 1250, 867, 1352, 2809, 486

Offset: 0

Paul Barry, Apr 03 2007

A divisibility sequence, that is, if n divides m then a(n) divides a(m). - Peter Bala, Feb 27 2019

G. C. Greubel, Table of n, a(n) for n = 0..1000
Peter Bala, A note on the sequence of numerators of a rational function , Feb 2019.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,3,0,0,0,0,0,-3,0,0,0,0,0,1).

Cf. A026741, A051176, A129196, A129197 (numerators), A060789.

[Denominator(3*(3+(-1)^n)/(n+1)^2): n in [0..50]]; // G. C. Greubel, Oct 26 2017

A129202:=n->numer((n+1)/2)*numer((n+1)/3): seq(A129202(n), n=0..100); # Wesley Ivan Hurt, Jul 18 2014

Table[Numerator[(n + 1)/2] Numerator[(n + 1)/3], {n, 0, 100}] (* Wesley Ivan Hurt, Jul 18 2014 *)
LinearRecurrence[{0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 1}, {1, 2, 3, 8, 25, 6, 49, 32, 27, 50, 121, 24, 169, 98, 75, 128, 289, 54}, 60] (* Harvey P. Dale, Nov 20 2016 *)

for(n=0,50, print1(denominator(3*(3+(-1)^n)/(n+1)^2), ", ")) \\ G. C. Greubel, Oct 26 2017

a(n) = A129196(n)/(n+1).
(1/(2*Pi))*Integral_{t=0..2*Pi} exp(i*(n+1)*t)*((t-Pi)/i)^3 dt = (a(n)*Pi^2-A129203(n))/A129196(n), i=sqrt(-1).
a(n) = ( Numerator of (n+1)/2 ) * ( Numerator of (n+1)/3 ) = A026741(n+1) * A051176(n+1). - Wesley Ivan Hurt, Jul 18 2014
G.f.: -(x^16 +2*x^15 +3*x^14 +8*x^13 +25*x^12 +6*x^11 +46*x^10 +26*x^9 +18*x^8 +26*x^7 +46*x^6 +6*x^5 +25*x^4 +8*x^3 +3*x^2 +2*x +1) / ((x -1)^3*(x +1)^3*(x^2 -x +1)^3*(x^2 +x +1)^3). - Colin Barker, Jul 18 2014
a(n+18) = 3*a(n+12)-3*a(n+6)+a(n). - Robert Israel, Jul 18 2014
a(n) = 2*(n+1)^2 * (7-4*cos(2*Pi*(n+1)/3)) / (9*(3-(-1)^n)). - Vaclav Kotesovec, Jul 20 2014
From Peter Bala, Feb 27 2019: (Start)
The following remarks assume an offset of 1.
a(n) = n^2/gcd(n,6) = n*A060789(n).
a(n) = n^2/b(n), where b(n) is the purely periodic sequence [1,2,3,2,1,6,...] with period 6. Thus a(n) is a quasi-polynomial in n:
a(6*n+1) = (6*n + 1)^2;
a(6*n+2) = 2*(3*n + 1)^2;
a(6*n+3) = 3*(2*n + 1)^2;
a(6*n+4) = 2*(3*n + 2)^2;
a(6*n+5) = (6*n + 5)^2;
a(6*n)   = 6*n^2.
O.g.f.: F(x) - 2*F(x^2) - 6*F(x^3) + 12*F(x^6), where F(x) = x*(1 + x)/(1 - x)^3 is the generating function for the squares. (End)
Sum_{n>=0} 1/a(n) = 55*Pi^2/216. - Amiram Eldar, Sep 27 2022

More terms from Wesley Ivan Hurt, Jul 18 2014

A216917 Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 12, 3, 2, 1, 1, 60, 3, 2, 1, 1, 1, 60, 15, 4, 3, 2, 1, 1, 420, 15, 20, 3, 6, 1, 1, 1, 840, 105, 20, 15, 12, 1, 2, 1, 1, 2520, 105, 140, 15, 12, 1, 6, 1, 1, 1, 2520, 315, 280, 105, 12, 5, 12, 3, 2, 1, 1, 27720, 315, 280, 105, 84

Offset: 1

Peter Luschny, Oct 02 2012

T(N,n) is the least common multiple of all integers up to N that are relatively prime to n.

Replacing LCM in the definition with "product" gives the Gauss factorial A216919.

			   n | N=0 1 2 3  4  5  6   7   8    9   10
-----+-------------------------------------
   1 |   1 1 2 6 12 60 60 420 840 2520 2520
   2 |   1 1 1 3  3 15 15 105 105  315  315
   3 |   1 1 2 2  4 20 20 140 280  280  280
   4 |   1 1 1 3  3 15 15 105 105  315  315
   5 |   1 1 2 6 12 12 12  84 168  504  504
   6 |   1 1 1 1  1  5  5  35  35   35   35
   7 |   1 1 2 6 12 60 60  60 120  360  360
   8 |   1 1 1 3  3 15 15 105 105  315  315
   9 |   1 1 2 2  4 20 20 140 280  280  280
  10 |   1 1 1 3  3  3  3  21  21   63   63
  11 |   1 1 2 6 12 60 60 420 840 2520 2520
  12 |   1 1 1 1  1  5  5  35  35   35   35
  13 |   1 1 2 6 12 60 60 420 840 2520 2520

t[, 0] = 1; t[n, k_] := LCM @@ Select[Range[k], CoprimeQ[#, n]&]; Table[t[n - k + 1, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Jul 29 2013 *)

def A216917(N, n):
    return lcm([j for j in (1..N) if gcd(j, n) == 1])
for n in (1..13): [A216917(N,n) for N in (0..10)]

For n > 0:
A(n,1) = A003418(n);
A(n,2^k) = A217858(n) for k > 0;
A(n,3^k) = A128501(n-1) for k > 0;
A(2,n) = A000034(n);
A(3,n) = A129203(n-1);
A(4,n) = A129197(n-1);
A(n,n) = A038610(n);
A(floor(n/2),n) = A124443(n);
A(n,1)/A(n,n) = A064446(n);
A(n,1)/A(n,2) = A053644(n).

cp's OEIS Frontend

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

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A129196 a(n) = denominator(3*(3+(-1)^n)/(n+1)^3).

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A283393 a(n) = gcd(n^2-1, n^2+9).

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A129202 Denominator of 3*(3+(-1)^n) / (n+1)^2.

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A216917 Square array read by antidiagonals, T(N,n) = lcm{1<=j<=N, gcd(j,n)=1 | j} for N >= 0, n >= 1.

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