A368527 -id:A368527 - cp's OEIS frontend

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

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.

A368526 a(n) = Sum_{k=1..n} k^2 * n^k.

Original entry on oeis.org

0, 1, 18, 282, 4740, 89355, 1896846, 45050852, 1186829064, 34391135205, 1087928669410, 37322190255966, 1380461544684300, 54772368958008975, 2320775754168090870, 104596636848116060040, 4996700995031905899536, 252208510175779038669321

Offset: 0

Seiichi Manyama, Dec 28 2023

Cf. A062806, A303991, A368527.

Cf. A368524.

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

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

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

A368537 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^k.

Original entry on oeis.org

0, 1, 18, 309, 5828, 123230, 2913126, 76405854, 2205340936, 69523722855, 2377899710410, 87721897714891, 3472488925101516, 146833416409808492, 6605726035373765678, 315051237815279406540, 15879038919798268666896, 843348814519524716426685

Offset: 0

Seiichi Manyama, Dec 29 2023

Cf. A062806, A368536.

Cf. A368527.

a(n) = sum(k=1, n, binomial(k+2, 3)*n^k);

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

a(n) = n * (n^n * (n^6-7*n^4+5*n^3+12*n^2-11*n-6) + 6)/(6 * (n-1)^4) for n > 1.

cp's OEIS Frontend

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

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A368526 a(n) = Sum_{k=1..n} k^2 * n^k.

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Formula

A368537 a(n) = Sum_{k=1..n} binomial(k+2,3) * n^k.

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PARI

Formula