A031971 -id:A031971 - cp's OEIS frontend

A349490 Sum of the n-th powers of the first n odd numbers.

1, 10, 153, 3108, 79225, 2437006, 87922513, 3642188936, 170423429841, 8891285549650, 511800291063721, 32222868466588460, 2202868653419747209, 162509566498902542934, 12868118600594014094625, 1088626666242258867384848, 97994054039441765759931169

Offset: 1

Seiichi Manyama, Dec 09 2021

nonn

Sum of the k-th powers of the first n odd numbers: A000290 (k=1), A000447 (k=2), A002593 (k=3), A002309 (k=4), A002594 (k=5), A259322 (k=6).

Cf. A005408, A031971, A061787, A103438.

Table[Sum[(2*k-1)^n, {k,1,n}], {n,1,20}] (* Vaclav Kotesovec, Dec 09 2021 *)

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

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

a(n) ~ 2^n * n^n / (exp(1/2) - exp(-1/2)). - Vaclav Kotesovec, Dec 09 2021

A368270 a(n) = Sum_{k=0..n} 2^(n-k) * k^n.

Original entry on oeis.org

1, 1, 6, 47, 490, 6417, 101178, 1866139, 39425322, 938856053, 24883226698, 726510389607, 23169961642698, 801435579830329, 29884247978965146, 1195036047465095027, 51016725208899539626, 2315820594694418639325, 111384453953719146198762

Offset: 0

Seiichi Manyama, Dec 25 2023

Cf. A031971, A120485, A349963.

PARI
```
a(n) = sum(k=0, n, 2^(n-k)*k^n);
```

a(n) ~ A309419 * n^n. - Vaclav Kotesovec, Dec 26 2023

A368492 a(n) = Sum_{k=0..n} k^(n+k).

Original entry on oeis.org

1, 1, 17, 762, 67788, 10034459, 2226678977, 691532161356, 286302140043208, 152380144397396525, 101369101598593980737, 82415052812980817676790, 80402388030057940015721524, 92697243086459794077519471767

Offset: 0

Seiichi Manyama, Dec 27 2023

Cf. A026898, A031971.

PARI
```
a(n) = sum(k=0, n, k^(n+k));
```

G.f.: Sum_{k>=0} (k^2*x)^k / (1 - k*x).

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

cp's OEIS Frontend

A349490 Sum of the n-th powers of the first n odd numbers.

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Mathematica

PARI

Formula

A368270 a(n) = Sum_{k=0..n} 2^(n-k) * k^n.

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Author

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Programs

PARI

Formula

A368492 a(n) = Sum_{k=0..n} k^(n+k).

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Author

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Formula