cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A368505 a(n) = Sum_{k=0..n} n^(n-k) * k^n.

Original entry on oeis.org

1, 1, 6, 60, 900, 18945, 536046, 19750864, 923023368, 53456736825, 3758407535410, 315059660550756, 31005946105301388, 3535337355213234025, 461813139285801522870, 68448013078470849802560, 11415281142784979557384208, 2126537819755999634260404945
Offset: 0

Views

Author

Seiichi Manyama, Dec 27 2023

Keywords

Crossrefs

Main diagonal of A368504.
Cf. A303991, A349836.

Programs

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

Formula

a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / (exp(n) * (log(n))^(n+1)). - Vaclav Kotesovec, Dec 28 2023

A303990 Triangle, read by rows: n^k * k^n, for n >= 1 and k = 1..n.

Original entry on oeis.org

1, 2, 16, 3, 72, 729, 4, 256, 5184, 65536, 5, 800, 30375, 640000, 9765625, 6, 2304, 157464, 5308416, 121500000, 2176782336, 7, 6272, 750141, 39337984, 1313046875, 32934190464, 678223072849, 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656
Offset: 1

Views

Author

Wolfdieter Lang, May 22 2018

Keywords

Comments

Due to the symmetry of n^k * k^n under the exchange n <-> k, it is sufficient to consider n >= 1, and k = 1..n.
For the array n^k * k^n, with n >= 0 and k >= 0, read by antidiagonals, see the triangle A062275.
Thanks go to S. Heinemeyer for leading me to look at this.
The row sums give A303991.

Examples

			The triangle T(n, k) begins:
======================================================================
n\k |  1    2      3        4          5           6            7  ...
----+-----------------------------------------------------------------
1:  |  1
2:  |  2   16
3:  |  3   72    729
4:  |  4  256   5184    65536
5:  |  5  800  30375   640000    9765625
6:  |  6 2304 157464  5308416  121500000  2176782336
7:  |  7 6272 750141 39337984 1313046875 32934190464 678223072849
...
row n=8: 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656;
row n=9: 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121;
row n=10: 10, 102400, 59049000, 10485760000, 976562500000, 60466176000000, 2824752490000000, 107374182400000000, 3486784401000000000, 100000000000000000000;
...

Crossrefs

Cf. A062275, A303991.

Programs

  • Magma
    /* As triangle */ [[n^k*k^n: k in [1..n]]: n in [1.. 15]]; // Vincenzo Librandi, May 23 2018
  • Mathematica
    Table[n^k k^n, {n, 10}, {k, n}] //Flatten (* Vincenzo Librandi, May 23 2018 *)
  • PARI
    T(n, k) = n^k * k^n;
tabl(nn) = for (n=1, nn, for (k=1, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, May 25 2018

Formula

T(n, k) = n^k * k^n, for n >= 1, k = 1..n.

A368486 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^j * j^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 3, 1, 0, 3, 18, 6, 1, 0, 4, 75, 90, 10, 1, 0, 5, 260, 804, 346, 15, 1, 0, 6, 805, 5444, 5988, 1146, 21, 1, 0, 7, 2310, 31180, 70980, 36363, 3450, 28, 1, 0, 8, 6279, 159774, 671180, 710980, 193827, 9722, 36, 1, 0, 9, 16392, 756420, 5468190, 10436805, 6019396, 943968, 26106, 45, 1
Offset: 0

Views

Author

Seiichi Manyama, Dec 26 2023

Keywords

Examples

			Square array begins:
  1,  0,    0,     0,      0,        0, ...
  1,  1,    2,     3,      4,        5, ...
  1,  3,   18,    75,    260,      805, ...
  1,  6,   90,   804,   5444,    31180, ...
  1, 10,  346,  5988,  70980,   671180, ...
  1, 15, 1146, 36363, 710980, 10436805, ...

Links

Crossrefs

Columns k=0..3 give A000012, A000217, A036800, A343808.
Main diagonal gives A303991.
Cf. A173018, A303990, A368479.

Programs

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

Formula

G.f. of column k: k*x*A_k(k*x)/((1-x) * (1-k*x)^(k+1)), where A_n(x) are the Eulerian polynomials for k > 0.

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

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Crossrefs

Cf. A062806, A303991, A368527.
Cf. A368524.

Programs

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

Formula

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.

A368527 a(n) = Sum_{k=1..n} k^3 * n^k.

Original entry on oeis.org

0, 1, 34, 804, 18244, 434205, 11138766, 310151632, 9370253320, 306232628625, 10783859167810, 407523041660196, 16461877678462668, 708207095198943613, 32338800248010936694, 1562509380160144645440, 79657105206246202521616
Offset: 0

Views

Author

Seiichi Manyama, Dec 28 2023

Keywords

Crossrefs

Cf. A062806, A303991, A368526.
Cf. A368525.

Programs

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

Formula

a(n) = [x^n] n*x * (1+4*n*x+(n*x)^2)/((1-x) * (1-n*x)^4).
a(n) = n * (n^n * (n^6-3*n^5+8*n^3-4*n^2-7*n-1) + n^2 + 4*n + 1)/(n-1)^4 for n > 1.

A351340 a(n) = Sum_{k=0..n} n^k * k^(n-k).

Original entry on oeis.org

1, 1, 6, 48, 516, 6955, 112686, 2132634, 46167560, 1125116901, 30481672610, 908760877244, 29565986232396, 1042354163621927, 39584173937284438, 1610922147768721590, 69940319175066857488, 3226793787576474492657, 157649292247463953189578
Offset: 0

Views

Author

Seiichi Manyama, Feb 08 2022

Keywords

Links

Crossrefs

Main diagonal of A351339.
Cf. A026898, A031973, A155956, A303991.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[n^k * k^(n - k), {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 08 2022 *)
  • PARI
    a(n) = sum(k=0, n, n^k*k^(n-k));

Formula

a(n) = [x^n] Sum_{k>=0} (n*x)^k/(1 - k*x).
a(n) ~ c * n^(n + 1/2), where c = sqrt(Pi)/2. - Vaclav Kotesovec, Feb 09 2022
Showing 1-6 of 6 results.

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