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A336203
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} 2^j * binomial(n,j)^k.
1
1, 1, 3, 1, 3, 7, 1, 3, 9, 15, 1, 3, 13, 27, 31, 1, 3, 21, 63, 81, 63, 1, 3, 37, 171, 321, 243, 127, 1, 3, 69, 495, 1521, 1683, 729, 255, 1, 3, 133, 1467, 7761, 14283, 8989, 2187, 511, 1, 3, 261, 4383, 41361, 131283, 138909, 48639, 6561, 1023, 1, 3, 517, 13131, 227601, 1256283, 2336629, 1385163, 265729, 19683, 2047
OFFSET
0,3
COMMENTS
Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - 2 * Product_{j=1..k} x_j) for k>0.
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
3, 3, 3, 3, 3, 3, ...
7, 9, 13, 21, 37, 69, ...
15, 27, 63, 171, 495, 1467, ...
31, 81, 321, 1521, 7761, 41361, ...
63, 243, 1683, 14283, 131283, 1256283, ...
MATHEMATICA
T[n_, k_] := Sum[2^j * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)
CROSSREFS
Columns k=0-4 give: A000225(n+1), A000244, A001850, A206178, A216696.
Main diagonal gives A336204.
Cf. A309010.
Sequence in context: A259325 A094250 A208517 * A209566 A208916 A209766
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Jul 11 2020
STATUS
approved