A368488 -id:A368488 - cp's OEIS frontend

A368487 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^j * binomial(j+k-1,j).

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 10, 17, 4, 1, 1, 17, 64, 49, 5, 1, 1, 26, 177, 334, 129, 6, 1, 1, 37, 401, 1457, 1549, 321, 7, 1, 1, 50, 793, 4776, 10417, 6652, 769, 8, 1, 1, 65, 1422, 12889, 48526, 67761, 27064, 1793, 9, 1, 1, 82, 2369, 30234, 176185, 442276, 411825, 105796, 4097, 10, 1

Offset: 0

Seiichi Manyama, Dec 26 2023

			Square array begins:
  1, 1,   1,    1,     1,      1, ...
  1, 2,   5,   10,    17,     26, ...
  1, 3,  17,   64,   177,    401, ...
  1, 4,  49,  334,  1457,   4776, ...
  1, 5, 129, 1549, 10417,  48526, ...
  1, 6, 321, 6652, 67761, 442276, ...

Columns k=0..3 give A000012, A000027(n+1), A000337(n+1), A367591.
Main diagonal gives A368488.
Cf. A119258, A368486.

T(n, k) = sum(j=0, n, k^j*binomial(j+k-1, j));

G.f. of column k: 1/((1-x) * (1-k*x)^k).

A368489 a(n) = Sum_{k=0..n} n^k * binomial(k+n,k).

Original entry on oeis.org

1, 3, 31, 643, 20421, 873806, 46994011, 3042431715, 230249448841, 19940350062394, 1944516598602711, 210829412453667998, 25156743053019602701, 3275876521195372322892, 462262670054775645538099, 70264375447526610838701091

Offset: 0

Seiichi Manyama, Dec 27 2023

Cf. A001700, A026641, A368488.

a(n) = sum(k=0, n, n^k*binomial(k+n, k));

a(n) = [x^n] 1/((1-x) * (1-n*x)^(n+1)).

a(n) ~ 4^n * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, Dec 27 2023

cp's OEIS Frontend

A368487 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^j * binomial(j+k-1,j).

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