A247239 Array a(n,m) = ((n+2)/2)^m*Sum_{k=1..n+1} 1/sin(k*Pi/(n+2))^(2m), n>=0, k>=0, read by ascending antidiagonals.

1, 2, 1, 3, 4, 1, 4, 10, 8, 1, 5, 20, 36, 16, 1, 6, 35, 120, 136, 32, 1, 7, 56, 329, 800, 528, 64, 1, 8, 84, 784, 3611, 5600, 2080, 128, 1, 9, 120, 1680, 13328, 42065, 40000, 8256, 256, 1, 10, 165, 3312, 42048, 241472, 499955, 288000, 32896, 512, 1

Offset

0,2

Comments

Unexpectedly, it is conjectured (proof wanted) that the expression ((n+2)/2)^m * Sum_{k=1..n+1} 1/sin(k*Pi/(n+2))^(2m), n>=0, k>=0, always gives an integer.

For example, a(3,1) = (5/2)*(1/sin(Pi/5)^2 + 1/sin((2*Pi)/5)^2 + 1/sin((3*Pi)/5)^2 + 1/sin((4*Pi)/5)^2) = (5/2)*(2/(5/8 - sqrt(5)/8) + 2/(5/8 + sqrt(5)/8)), which simplifies to 20.

Links

Table of n, a(n) for n=0..54.

I. M. Gessel, Generating functions and generalized Dedekind sums, Electron. J. Combin.4 (1997), no. 2, Research Paper 11, 17 pp.

Les Mathematiques, Somme des 1/sin^2, Sketch of a proof [in French].

MilesB, How to prove this sum involving powers of cosec is an integer?, MathOverflow 444094.

R. P. Stanley, Invariants of finite groups and their applications to combinatorics, Bull. Amer. Math. Soc. 1 (1979), 475-511.

Formula

First formulas for rows:

a(0,m) = 1.

a(1,m) = 2^(m + 1).

a(2,m) = 2^m + 2^(2*m + 1).

a(3,m) = 2*((5 - sqrt(5))^m + (5 + sqrt(5))^m).

a(4,m) = 2^(2*m + 1) + 3^m + 2^(2*m + 1)*3^m.

First formulas for columns:

a(n,0) = n + 1.

a(n,1) = (n + 1)*(n + 2)*(n + 3)/6.

a(n,2) = coefficient of x^n in the expansion of (1 - x^4)/(1 - x)^8.

Let b(N,m) be (N/2)^m times the coefficient of x^(2*m) in 1-N*x*cot(N*arcsin(x))/ sqrt(1-x^2). Then for m>0, a(n,m) = b(n+2,m). - Ira M. Gessel, Apr 04 2023

Example

Array a(n,m) begins:
  1,  1,   1,    1,     1,      1,       1,        1, ... 1 (A000012)
  2,  4,   8,   16,    32,     64,     128,      256, ... 2^(m+1) (A000079)
  3, 10,  36,  136,   528,   2080,    8256,    32896, ... A007582
  4, 20, 120,  800,  5600,  40000,  288000,  2080000, ... A093123
  5, 35, 329, 3611, 42065, 499955, 5980889, 71698571, ... not in the OEIS
  ...
1st column is n+1 (A000027).
2nd column is A000292.
3rd column is not in the OEIS.

Mathematica

a[n_, m_] := ((n + 2)/2)^m*Sum[1/Sin[k*(Pi/(n + 2))]^(2*m), {k, 1, n + 1}]; Table[a[n - m, m] // FullSimplify, {n, 0, 10}, {m, 0, n}] // Flatten

Prog

(PARI) a(n, m)={t=Pi/(n+2); u=1+n/2; round(sum(k=1, n+1, (u/sin(k*t)^2)^m))} \\ M. F. Hasler, Dec 03 2014

Crossrefs

Cf. A000079, A000292, A007582, A093123.

Sequence in context: A078812 A104711 A133112 * A198060 A327084 A159856

Adjacent sequences: A247236 A247237 A247238 * A247240 A247241 A247242

Keyword

nonn,tabl

Author

Jean-François Alcover, Nov 28 2014

Status

approved