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A376499 Array read by ascending antidiagonals: A(n,k) = A376484/(2*n+1).

%I A376499 #53 Jun 01 2025 10:07:00
%S A376499 1,1,3,1,3,9,1,3,10,27,1,3,10,35,81,1,3,10,35,125,243,1,3,10,35,126,
%T A376499 450,729,1,3,10,35,126,462,1625,2187,1,3,10,35,126,462,1715,5875,6561,
%U A376499 1,3,10,35,126,462,1716,6419,21250,19683,1,3,10,35,126,462,1716,6435,24157,76875,59049
%N A376499 Array read by ascending antidiagonals: A(n,k) = A376484/(2*n+1).
%C A376499 It is only a conjecture that the A(n,k) are always integers.
%C A376499 Values repeated as a staircase for all A(n+x,2*n) (x > 0 and are equal to A(n,2*n)).
%F A376499 A(n,k) = A376484(n,k)/(2*n+1)
%e A376499 First ten rows start as follows:
%e A376499   1 3  9 27  81 243  729 2187  6561 19683  59049  177147  531441  1594323  4782969
%e A376499   1 3 10 35 125 450 1625 5875 21250 76875 278125 1006250 3640625 13171875 47656250
%e A376499   1 3 10 35 126 462 1715 6419 24157 91238 345401 1309574 4970070 18874261 71705865
%e A376499   1 3 10 35 126 462 1716 6435 24309 92358 352485 1350054 5185350 19960020 76964985
%e A376499   1 3 10 35 126 462 1716 6435 24310 92378 352715 1352054 5199975 20055024 77531355
%e A376499   1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200299 20058272 77558325
%e A376499   1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558759
%e A376499   1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
%e A376499   1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
%e A376499   1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
%o A376499 (C++) double gen(int n, int m) {
%o A376499     double s = 0, d = 1;
%o A376499     for(int i = 1; i <= m; i++) d *= 4;
%o A376499     for(int i = 1; i <= n; i++) {
%o A376499         double v = 1;
%o A376499         for(int j = 1; j <= 2 * m; j++) v *= sin(2 * i * M_PI / (2 * n + 1));
%o A376499         s += v * d;
%o A376499     }
%o A376499     return s / (2 * n + 1);
%o A376499 }
%Y A376499 All of these are conjectures. Rows: A000244, A081567, A122068. Columns: A000012, A000012 * 3, A095049 for n >= 20. A(1,k) = A000244, A(2,k) = A081567, A(3,k) = A122068 (First 3 rows of the array).A(n,1) = A(n,2) / 3 = A000012, A(n,3) = A095049 for n >= 20 (First 3 columns of the array). When k increases, the row of A(n,k) gets closer to A001700.
%K A376499 nonn,tabl
%O A376499 1,3
%A A376499 _Cheng-Jun Li_, Sep 25 2024