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A368504
Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * j^k.
1
1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 6, 6, 1, 0, 1, 11, 21, 10, 1, 0, 1, 20, 60, 58, 15, 1, 0, 1, 37, 161, 244, 141, 21, 1, 0, 1, 70, 428, 900, 857, 318, 28, 1, 0, 1, 135, 1149, 3164, 4225, 2787, 685, 36, 1, 0, 1, 264, 3132, 10990, 18945, 18196, 8704, 1434, 45, 1
OFFSET
0,9
FORMULA
G.f. of column k: x*A_k(x)/((1-k*x) * (1-x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.
T(0,k) = 0^k; T(n,k) = k*T(n-1,k) + n^k.
EXAMPLE
Square array begins:
1, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 3, 6, 11, 20, 37, 70, ...
1, 6, 21, 60, 161, 428, 1149, ...
1, 10, 58, 244, 900, 3164, 10990, ...
1, 15, 141, 857, 4225, 18945, 81565, ...
1, 21, 318, 2787, 18196, 102501, 536046, ...
PROG
(PARI) T(n, k) = sum(j=0, n, k^(n-j)*j^k);
CROSSREFS
Columns k=0..5 give A000012, A000217, A047520, A066999, A067534, A218376.
Main diagonal gives A368505.
Cf. A368486.
Sequence in context: A122935 A131198 A090181 * A256551 A144417 A085791
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Dec 27 2023
STATUS
approved