OFFSET
0,4
COMMENTS
It is only a conjecture that A(n,k) is always an integer.
It appears that A(n,k) is divisible by 2*n+1 when n, k are positive integers.
EXAMPLE
For n = 0 to 10 and k = 0 to 10, A(n, k) shows as below :
0 0 0 0 0 0 0 0 0 0 0
1 3 9 27 81 243 729 2187 6561 19683 59049
2 5 15 50 175 625 2250 8125 29375 106250 384375
3 7 21 70 245 882 3234 12005 44933 169099 638666
4 9 27 90 315 1134 4158 15444 57915 218781 831222
5 11 33 110 385 1386 5082 18876 70785 267410 1016158
6 13 39 130 455 1638 6006 22308 83655 316030 1200914
7 15 45 150 525 1890 6930 25740 96525 364650 1385670
8 17 51 170 595 2142 7854 29172 109395 413270 1570426
9 19 57 190 665 2394 8778 32604 122265 461890 1755182
10 21 63 210 735 2646 9702 36036 135135 510510 1939938
PROG
(C++) double gen(int n, int m) {
double s = 0, d = 1;
for(int i = 1; i <= m; i++) d *= 4;
for(int i = 1; i <= n; i++) {
double v = 1;
for(int j = 1; j <= 2 * m; j++) v *= sin(2 * i * M_PI / (2 * n + 1));
s += v * d;
}
return s;
}
(PARI) A(n, k) = 4^k*sum(j=1, n, (sin(2*j*Pi/(2*n+1)))^(2*k), x=0)
CROSSREFS
Conjectures: This array is related to existing sequebces as follows: (Start)
Main Diagonals: A033876.
A(0,k) = A000004, A(1,k) = A000244, A(2,k) = A020876, A(3,k) = (-1)^k * A322459 (First 4 rows of the array).
A(n,0) = A001477(n > 0), A(n,1) = A005408(n > 0), A(n,2) = A016945(n > 0) (First 3 columns of the array).
A(n,n) = A033876(n > 0) (Main diagonal from top left corner). (End)
KEYWORD
nonn,tabl
AUTHOR
Cheng-Jun Li, Sep 24 2024
STATUS
approved