A062075 -id:A062075 - cp's OEIS frontend

A098803 a(n) = n^7 * 7^n.

0, 7, 6272, 750141, 39337984, 1313046875, 32934190464, 678223072849, 12089663946752, 193010051319183, 2824752490000000, 38532504363714053, 495958345459089408, 6079641716636816419, 71493870602660352896

Offset: 0

Parthasarathy Nambi, Oct 05 2004

			a(1) = 1^7 * 7^1 = 7.
a(2) = 2^7 * 7^2 = 6272.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (56,-1372,19208,-168070,941192,-3294172,6588344,-5764801).

Cf. A007758, A062074, A062075.

[7^n*n^7: n in [0..20]]; // Vincenzo Librandi, Oct 27 2011

Table[n^7*7^n, {n, 1, 20}] (* Stefan Steinerberger, Mar 06 2006 *)

a(n)=n^7*7^n \\ Charles R Greathouse IV, Oct 07 2015

G.f.: 7*x*(117649*x^6 +2016840*x^5 +2859591*x^4 +828688*x^3 +58359*x^2 +840*x +1) / (7*x -1)^8. - Colin Barker, Apr 30 2013

More terms from Stefan Steinerberger, Mar 06 2006

Offset changed from 1 to 0 by Vincenzo Librandi, Oct 27 2011

A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 16, 3, 0, 0, 4, 72, 72, 4, 0, 0, 5, 256, 729, 256, 5, 0, 0, 6, 800, 5184, 5184, 800, 6, 0, 0, 7, 2304, 30375, 65536, 30375, 2304, 7, 0, 0, 8, 6272, 157464, 640000, 640000, 157464, 6272, 8, 0, 0, 9, 16384, 750141, 5308416, 9765625

Offset: 0

Henry Bottomley, Jul 02 2001

Here 0^0 is defined to be 1. - Wolfdieter Lang, May 27 2018

			A(3, 2) = 3^2 * 2^3 = 9*8 = 72.
The array A(n, k) begins:
n\k 0 1   2   3    4     5      6      7       8        9       10 ...
0:  1 0   0   0    0     0      0      0       0        0        0 ...
1:  0 1   2   3    4     5      6      7       8        9       10 ...
2:  0 2  16  72  256   800   2304   6272   16384    41472   102400 ...
3:  0 3  72 729 5184 30375 157464 750141 3359232 14348907 59049000 ...
...
The triangle T(n, k) begins:
n\k  0  1    2      3      4      5      6    7  8  9 ...
0:   1
1:   0  0
2:   0  1    0
3:   0  2    2      0
4:   0  3   16      3      0
5:   0  4   72     72      4      0
6:   0  5  256    729    256      5      0
7:   0  6  800   5184   5184    800      6    0
8:   0  7 2304  30375  65536  30375   2304    7  0
9:   0  8 6272 157464 640000 640000 157464 6272  8  0
... - _Wolfdieter Lang_, May 22 2018

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)

Columns and rows of A, or columns and diagonals of T, include A000007, A001477, A007758, A062074, A062075 etc. Diagonals of A include A062206, A051443, A051490. Sum of rows of T are A062817(n), for n >= 1

Cf. A055651, A055652, A303990.

{{1}}~Join~Table[(#^k k^#) &[n - k], {n, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, May 24 2018 *)

t1(n)=n-binomial(round(sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
a(n)=t1(n)^t2(n)*t2(n)^t1(n) \\ Eric Chen, Jun 09 2018

From Wolfdieter Lang, May 22 2018: (Start)
As a sequence: a(n) = A003992(n)*A004248(n).
As a triangle: T(n, k) = (n-k)^k * k^(n-k), for n >= 1 and k = 1..n. (End)

A146748 Numbers of the form n^k * k^n, where n,k > 1.

Original entry on oeis.org

16, 72, 256, 729, 800, 2304, 5184, 6272, 16384, 30375, 41472, 65536, 102400, 157464, 247808, 589824, 640000, 750141, 1384448, 3211264, 3359232, 5308416, 7372800, 9765625, 14348907, 16777216, 37879808, 39337984, 59049000, 84934656

Offset: 1

Howard Berman (howard_berman(AT)hotmail.com), Nov 01 2008

nonn

			2^2 * 2^2 = 16,
2^3 * 3^2 = 72.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007758, A062074, A062075, A062275.

N:= 10^20: # for terms <= N
S:= {}:
for n from 2 to ilog2(N) do
  for k from n do
    v:= n^k * k^n;
    if v > N then break fi;
    S:= S union {v};
od od:
sort(convert(S,list)); # Robert Israel, Oct 31 2023

A198404 8^n*n^8.

Original entry on oeis.org

0, 8, 16384, 3359232, 268435456, 12800000000, 440301256704, 12089663946752, 281474976710656, 5777633090469888, 107374182400000000, 1841328767004311552, 29548117155177824256, 448452706436800053248, 6490588908866265677824, 90173697372979200000000

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (72,-2304,43008,-516096,4128768,-22020096,75497472,-150994944,134217728).

Cf. A007758, A062074, A062075, A097206, A098880.

Magma
```
[8^n*n^8: n in [0..20]]
```

Table[8^n n^8,{n,0,20}] (* or *) LinearRecurrence[{72,-2304,43008,-516096,4128768,-22020096,75497472,-150994944,134217728},{0,8,16384,3359232,268435456,12800000000,440301256704,12089663946752,281474976710656},20] (* Harvey P. Dale, Apr 28 2018 *)

a(n)=8^n*n^8 \\ Charles R Greathouse IV, Jul 06 2017

G.f.: -8*x*(8*x +1)*(262144*x^6 +8060928*x^5 +16576512*x^4 +5924864*x^3 +259008*x^2 +1968*x +1) / (8*x -1)^9. - Colin Barker, Apr 30 2013

A198478 a(n) = 9^n * n^9.

Original entry on oeis.org

0, 9, 41472, 14348907, 1719926784, 115330078125, 5355700839936, 193010051319183, 5777633090469888, 150094635296999121, 3486784401000000000, 73994897046174912819, 1457274373159131021312, 26955214582765006137717

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Cf. A001017, A001019.

Cf. A007758, A062074, A062075, A097206, A098803, A098880.

Magma
```
[9^n*n^9: n in [0..20]]
```

Table[9^n*n^9, {n, 0, 20}] (* G. C. Greubel, May 17 2022 *)

[9^n*n^9 for n in (0..20)] # G. C. Greubel, May 17 2022

G.f.: 9*x*(1 + 4518*x + 1183248*x^2 + 64322586*x^3 + 1024762590*x^4 + 5210129466*x^5 + 7763290128*x^6 + 2401050438*x^7 + 43046721*x^8)/(1 - 9*x)^10. - Colin Barker, Apr 30 2013

a(n) = A001019(n)*A001017(n). - Michel Marcus, May 18 2022

A198479 a(n) = 10^n * n^10.

Original entry on oeis.org

0, 10, 102400, 59049000, 10485760000, 976562500000, 60466176000000, 2824752490000000, 107374182400000000, 3486784401000000000, 100000000000000000000, 2593742460100000000000, 61917364224000000000000

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..900

Cf. A008454, A011557.

Cf. A007758, A062074, A062075, A097206, A098803, A098880.

Magma
```
[10^n*n^10: n in [0..20]]
```

Table[10^n*n^10, {n,0,20}] (* G. C. Greubel, May 17 2022 *)

[10^n*n^10 for n in (0..20)] # G. C. Greubel, May 17 2022

G.f.: 10*x*(1 + 10*x)*(1 + 10120*x + 4682800*x^2 + 408364000*x^3 + 9019900000*x^4 + 40836400000*x^5 + 46828000000*x^6 + 10120000000*x^7 + 100000000*x^8)/ (1-10*x)^11. - Colin Barker, May 01 2013

a(n) = A011557(n)*A008454(n). - Michel Marcus, May 18 2022

A198402 a(n) = 5^n * n^5.

Original entry on oeis.org

0, 5, 800, 30375, 640000, 9765625, 121500000, 1313046875, 12800000000, 115330078125, 976562500000, 7863818359375, 60750000000000, 453238525390625, 3282617187500000, 23174285888671875, 160000000000000000, 1083264923095703125

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (30,-375,2500,-9375,18750,-15625).

Cf. A000351, A000584, A097206.

Sequences of the form n^m*m^n: A001477 (m=1), A007758 (m=2), A062074 (m=3), A062075 (m=4), this sequence (m=5), A198403 (m=6), A098803 (m=7), A198404 (m=8), A198478 (m=9), A198479 (m=10), A098880 (m=11).

Magma
```
[5^n*n^5: n in [0..20]]
```

With[{m = 5}, Table[n^m*m^n, {n, 0, 30}]] (* G. C. Greubel, May 18 2022 *)

a(n)=5^n*n^5 \\ Charles R Greathouse IV, Jul 06 2017

m=5; [n^m*m^n for n in (0..30)] # G. C. Greubel, May 18 2022

G.f.: 5*x*(1 + 130*x + 1650*x^2 + 3250*x^3 + 625*x^4)/(1-5*x)^6. - Colin Barker, Apr 30 2013
E.g.f.: 5*x*(1 + 75*x + 625*x^2 + 1250*x^3 + 625*x^4)*exp(5*x). - G. C. Greubel, May 18 2022
a(n) = A000351(n)*A000584(n). - Michel Marcus, May 19 2022

A198403 a(n) = 6^n * n^6.

Original entry on oeis.org

0, 6, 2304, 157464, 5308416, 121500000, 2176782336, 32934190464, 440301256704, 5355700839936, 60466176000000, 642717115324416, 6499837226778624, 63041475422674944, 590045794670739456, 5355700839936000000

Offset: 0

Vincenzo Librandi, Oct 27 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (42,-756,7560,-45360,163296,-326592,279936).

Cf. A000400, A001014.

Sequences of the form n^m*m^n: A001477 (m=1), A007758 (m=2), A062074 (m=3), A062075 (m=4), A198402 (m=5), this sequence (m=6), A098803 (m=7), A198404 (m=8), A198478 (m=9), A198479 (m=10), A098880 (m=11).

Magma
```
[6^n*n^6: n in [0..20]]
```

With[{m=6}, Table[n^m*m^n, {n,0,30}]] (* G. C. Greubel, May 18 2022 *)

a(n)=6^n*n^6 \\ Charles R Greathouse IV, Jul 06 2017

m=6; [n^m*m^n for n in (0..30)] # G. C. Greubel, May 18 2022

G.f.: 6*x*(1 + 336*x + 8856*x^2 + 12096*x^3 + 1296*x^4)/(1-6*x)^7. - Colin Barker, Apr 30 2013
E.g.f.: 6*x*(1 + 186*x + 3240*x^2 + 14040*x^3 + 19440*x^4 + 7776*x^5)*exp(6*x). - G. C. Greubel, May 18 2022
a(n) = A000400(n)*A001014(n). - Michel Marcus, May 19 2022
a(n) = 42*a(n-1) - 756*a(n-2) + 7560*a(n-3) - 45360*a(n-4) + 163296*a(n-5) - 326592*a(n-6) + 279936*a(n-7). - Wesley Ivan Hurt, Sep 04 2022

cp's OEIS Frontend

A098803 a(n) = n^7 * 7^n.

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A062275 Array A(n, k) = n^k * k^n, n, k >= 0, read by antidiagonals.

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A146748 Numbers of the form n^k * k^n, where n,k > 1.

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Maple

A198404 8^n*n^8.

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A198478 a(n) = 9^n * n^9.

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A198479 a(n) = 10^n * n^10.

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A198402 a(n) = 5^n * n^5.

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