A368505 -id:A368505 - cp's OEIS frontend

A303991 Row sums of triangle A303990.

1, 18, 804, 70980, 10436805, 2303750526, 712510404592, 294018013725192, 156070751204023425, 103597044789173411410, 84072367255899882570876, 81892130447332894817380044, 94289343231845338982163322837, 126676207083751543195799431746150, 196394200592428254386554058525461440

Offset: 1

Wolfdieter Lang, May 22 2018

Seiichi Manyama, Table of n, a(n) for n = 1..214

Cf. A303990, A368505.

Total /@ Table[n^k k^n, {n, 15}, {k, n}] (* Michael De Vlieger, May 24 2018 *)

a(n) = sum(k=1, n, n^k * k^n); \\ Michel Marcus, May 25 2018

a(n) = Sum_{k=1..n} A303990(n, k) = Sum_{k=1..n} n^k * k^n, for n >= 1.

a(n) ~ n^(2*n). - Vaclav Kotesovec, Feb 09 2022

A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

Original entry on oeis.org

1, 1, 5, 44, 564, 9665, 211025, 5686104, 184813048, 7118824417, 320295658577, 16626717667348, 985178854556524, 66005199079345025, 4958773228726876257, 414664315430994701616, 38344259607889223269168, 3898112616839310343827009

Offset: 0

Seiichi Manyama, Dec 03 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..299

Cf. A031971, A349883.

Cf. A249459, A349884, A368505.

Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 04 2021 *)

a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));

my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^2*x)))

a(n) = Sum_{k=0..n} k^(2*n-k).

a(n) ~ sqrt(Pi) * 2^(1 + 2*n - 2*n/LambertW(2*exp(1)*n)) * (n/LambertW(2*exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A368524 a(n) = Sum_{k=1..n} k^2 * n^(n-k).

Original entry on oeis.org

0, 1, 6, 30, 180, 1455, 15666, 213500, 3521736, 68101245, 1508916310, 37661140506, 1045012524348, 31900040161899, 1062139933257690, 38299757176168440, 1486670929792295696, 61800664096000744569, 2738952078516469743678, 128909373997071187219990

Offset: 0

Seiichi Manyama, Dec 28 2023

Cf. A062805, A368505, A368525.

Cf. A368526, A368534.

PARI
```
a(n) = sum(k=1, n, k^2*n^(n-k));
```

a(n) = [x^n] x * (1+x)/((1-n*x) * (1-x)^3).

a(n) = n * (n+1) * (n^n - n^2 + n - 1)/(n-1)^3 for n > 1.

A368525 a(n) = Sum_{k=1..n} k^3 * n^(n-k).

Original entry on oeis.org

0, 1, 10, 60, 364, 2745, 27246, 346864, 5422264, 100449225, 2149062490, 52097910876, 1410401518692, 42153624499441, 1378058477508454, 48900582823143360, 1871456346915007216, 76821658841556480753, 3366451935514051046802, 156839738363103277783900

Offset: 0

Seiichi Manyama, Dec 28 2023

Cf. A062805, A368505, A368524.

Cf. A368527.

PARI
```
a(n) = sum(k=1, n, k^3*n^(n-k));
```

a(n) = [x^n] x * (1+4*x+x^2)/((1-n*x) * (1-x)^4).

a(n) = n * (n^n * (n^2 + 4*n + 1) - n^5 - 3*n^2 - n - 1)/(n-1)^4 for n > 1.

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.

Original entry on oeis.org

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

Seiichi Manyama, Dec 27 2023

			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, ...

OEIS Wiki, Eulerian polynomials.

Columns k=0..5 give A000012, A000217, A047520, A066999, A067534, A218376.
Main diagonal gives A368505.
Cf. A368486.

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

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.

cp's OEIS Frontend

A303991 Row sums of triangle A303990.

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A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

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A368524 a(n) = Sum_{k=1..n} k^2 * n^(n-k).

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A368525 a(n) = Sum_{k=1..n} k^3 * n^(n-k).

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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.

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