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A376484
Array read by ascending antidiagonals: A(n,k)=4^k*Sum_{j=1..n} sin(2*j*Pi/(2*n+1))^(2*k).
1
0, 1, 0, 2, 3, 0, 3, 5, 9, 0, 4, 7, 15, 27, 0, 5, 9, 21, 50, 81, 0, 6, 11, 27, 70, 175, 243, 0, 7, 13, 33, 90, 245, 625, 729, 0, 8, 15, 39, 110, 315, 882, 2250, 2187, 0, 9, 17, 45, 130, 385, 1134, 3234, 8125, 6561, 0, 10, 19, 51, 150, 455, 1386, 4158, 12005, 29375, 19683, 0
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)
Rows: A000004, A000244, A020876, A322459 (with alternate signs).
Columns: A001477, A005408, A016945.
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)
Sequence in context: A047773 A279416 A331781 * A187988 A035549 A329970
KEYWORD
nonn,tabl
AUTHOR
Cheng-Jun Li, Sep 24 2024
STATUS
approved