A368487 -id:A368487 - cp's OEIS frontend

A368488 a(n) = Sum_{k=0..n} n^k * binomial(k+n-1,k).

1, 2, 17, 334, 10417, 442276, 23690809, 1530206742, 115636017473, 10004657077468, 974950612575601, 105653682110368492, 12602144701834193521, 1640558582759557298696, 231448351542446473323113, 35173958220088874039434726, 5728588740444710703061240065

Offset: 0

Seiichi Manyama, Dec 26 2023

Main diagonal of A368487.

Cf. A000984, A072547.

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

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

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

A368506 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * binomial(j+k-1,j).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 4, 3, 0, 1, 6, 11, 4, 0, 1, 8, 24, 26, 5, 0, 1, 10, 42, 82, 57, 6, 0, 1, 12, 65, 188, 261, 120, 7, 0, 1, 14, 93, 360, 787, 804, 247, 8, 0, 1, 16, 126, 614, 1870, 3204, 2440, 502, 9, 0, 1, 18, 164, 966, 3810, 9476, 12900, 7356, 1013, 10, 0

Offset: 0

Seiichi Manyama, Dec 27 2023

			Square array begins:
  1, 1,   1,    1,     1,     1,      1, ...
  0, 2,   4,    6,     8,    10,     12, ...
  0, 3,  11,   24,    42,    65,     93, ...
  0, 4,  26,   82,   188,   360,    614, ...
  0, 5,  57,  261,   787,  1870,   3810, ...
  0, 6, 120,  804,  3204,  9476,  23112, ...
  0, 7, 247, 2440, 12900, 47590, 139134, ...

Columns k=0..3 give A000007, A000027(n+1), A125128(n+1), A052150.
Main diagonal gives A293574.
Cf. A008949, A368487.

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

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

cp's OEIS Frontend

A368488 a(n) = Sum_{k=0..n} n^k * binomial(k+n-1,k).

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