A368524 -id:A368524 - 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.

A368529 a(n) = Sum_{k=1..n} k^2 * 4^(n-k).

Original entry on oeis.org

0, 1, 8, 41, 180, 745, 3016, 12113, 48516, 194145, 776680, 3106841, 12427508, 49710201, 198841000, 795364225, 3181457156, 12725828913, 50903315976, 203613264265, 814453057460, 3257812230281, 13031248921608, 52124995686961, 208499982748420, 833999930994305

Offset: 0

Seiichi Manyama, Dec 28 2023

Paolo Xausa, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-15,13,-4).

Cf. A000290, A000330, A047520, A368528.
Cf. A052161, A368524.
Cf. A014825, A368530.

LinearRecurrence[{7, -15, 13, -4}, {0, 1, 8, 41}, 30] (* Paolo Xausa, Jan 29 2024 *)

PARI
```
a(n) = sum(k=1, n, k^2*4^(n-k));
```

G.f.: x * (1+x)/((1-4*x) * (1-x)^3).
a(n) = 7*a(n-1) - 15*a(n-2) + 13*a(n-3) - 4*a(n-4).
a(n) = A052161(n-1) + A052161(n-2) for n > 1.
a(n) = (5*4^(n+1) - (9*n^2 + 24*n + 20))/27.
a(0) = 0; a(n) = 4*a(n-1) + n^2.

A368528 a(n) = Sum_{k=1..n} k^2 * 3^(n-k).

Original entry on oeis.org

0, 1, 7, 30, 106, 343, 1065, 3244, 9796, 29469, 88507, 265642, 797070, 2391379, 7174333, 21523224, 64569928, 193710073, 581130543, 1743391990, 5230176370, 15690529551, 47071589137, 141214767940, 423644304396, 1270932913813, 3812798742115

Offset: 0

Seiichi Manyama, Dec 28 2023

Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Cf. A000290, A000330, A047520, A368529.

Cf. A052150, A368524.

PARI
```
a(n) = sum(k=1, n, k^2*3^(n-k));
```

G.f.: x * (1+x)/((1-3*x) * (1-x)^3).
a(n) = 6*a(n-1) - 12*a(n-2) + 10*a(n-3) - 3*a(n-4).
a(n) = A052150(n-1) + A052150(n-2) for n > 1.
a(n) = (3^(n+1) - (n^2 + 3*n + 3))/2.
a(0) = 0; a(n) = 3*a(n-1) + n^2.

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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A368529 a(n) = Sum_{k=1..n} k^2 * 4^(n-k).

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Mathematica

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A368528 a(n) = Sum_{k=1..n} k^2 * 3^(n-k).

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