A351279 -id:A351279 - cp's OEIS frontend

A351282 a(n) = Sum_{k=0..n} 3^k * k^(n-k).

1, 3, 12, 48, 201, 885, 4116, 20298, 106365, 592455, 3503532, 21946620, 145210305, 1011726417, 7400390052, 56668826118, 453116188821, 3774297532467, 32682069679548, 293632972911048, 2732593851548985, 26299137526992525, 261387306941467188, 2679392140776188706

Offset: 0

Vaclav Kotesovec, Feb 06 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..572

Cf. A003101, A351279.

Join[{1}, Table[Sum[3^k*k^(n-k), {k, 0, n}], {n, 1, 25}]]

a(n) = sum(k=0, n, 3^k*k^(n-k)); \\ Michel Marcus, Feb 06 2022

a(n) ~ sqrt(2*Pi/(1 + LambertW(exp(1)*n/3))) * n^(n + 1/2) * exp(n/LambertW(exp(1)*n/3) - n) / LambertW(exp(1)*n/3)^(n + 1/2).

G.f.: Sum_{k>=0} 3^k * x^k / (1 - k*x). - Ilya Gutkovskiy, Feb 06 2022

A351339 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is Sum_{j=0..n} k^j * j^(n-j).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 6, 4, 0, 1, 4, 12, 18, 9, 0, 1, 5, 20, 48, 58, 23, 0, 1, 6, 30, 100, 201, 202, 66, 0, 1, 7, 42, 180, 516, 885, 762, 210, 0, 1, 8, 56, 294, 1105, 2756, 4116, 3114, 733, 0, 1, 9, 72, 448, 2094, 6955, 15300, 20298, 13754, 2781, 0

Offset: 0

Seiichi Manyama, Feb 08 2022

			Square array begins:
  1,  1,   1,    1,     1,     1,      1, ...
  0,  1,   2,    3,     4,     5,      6, ...
  0,  2,   6,   12,    20,    30,     42, ...
  0,  4,  18,   48,   100,   180,    294, ...
  0,  9,  58,  201,   516,  1105,   2094, ...
  0, 23, 202,  885,  2756,  6955,  15198, ...
  0, 66, 762, 4116, 15300, 45030, 112686, ...

Columns k=0..3 give A000007, A026898(n-1), A351279, A351282.

Main diagonal gives A351340.

T[0, k_] := 1; T[n_, 0] = 0; T[n_, k_] := Sum[k^j * j^(n - j), {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 08 2022 *)

PARI
```
T(n, k) = sum(j=0, n, k^j*j^(n-j));
```

G.f. of column k: Sum_{j>=0} (k*x)^j/(1 - j*x).

cp's OEIS Frontend

A351282 a(n) = Sum_{k=0..n} 3^k * k^(n-k).

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