A376498 Array read by ascending antidiagonals: A(n, k) = 2^k*Sum_{j=1..n} cos((2*j - 1)*Pi/(2*n + 1))^k.
0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 1, 3, 1, 0, 5, 1, 5, 4, 1, 0, 6, 1, 7, 4, 7, 1, 0, 7, 1, 9, 4, 13, 11, 1, 0, 8, 1, 11, 4, 19, 16, 18, 1, 0, 9, 1, 13, 4, 25, 16, 38, 29, 1, 0, 10, 1, 15, 4, 31, 16, 58, 57, 47, 1, 0, 11, 1, 17, 4, 37, 16, 78, 64, 117, 76, 1, 0
Offset: 0
Examples
Array starts: [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... [A000004] [1] 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ... [A000012] [2] 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, ... [A000032] [3] 3, 1, 5, 4, 13, 16, 38, 57, 117, 193, 370, 639, ... [A096975] [4] 4, 1, 7, 4, 19, 16, 58, 64, 187, 247, 622, 925, ... [A094649] [5] 5, 1, 9, 4, 25, 16, 78, 64, 257, 256, 874, 1013, ... [A189234] [6] 6, 1, 11, 4, 31, 16, 98, 64, 327, 256, 1126, 1024, ... [A216605] [7] 7, 1, 13, 4, 37, 16, 118, 64, 397, 256, 1378, 1024, ... [8] 8, 1, 15, 4, 43, 16, 138, 64, 467, 256, 1630, 1024, ... [9] 9, 1, 17, 4, 49, 16, 158, 64, 537, 256, 1882, 1024, ...
Crossrefs
Programs
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PARI
A(n, k) = 2^k*sum(j=1, n, (cos((2*j-1)*Pi/(2*n+1)))^k, x=0)
Formula
A(n + k, 2*k - 1) = A(k, 2*k-1) = 4^(k-1).
Let P_n(x) be the polynomial: Sum_{k=0..n} x^k*A180870(n, k). Let R_n(x) be the polynomial Product_{k=0..n} x-Roots(P_n, k)^m. A(n, k) = abs([x^1] R_n(x))/2^(m*(n-1)), for n > 0. - Thomas Scheuerle, Oct 07 2024
Comments