A062206 -id:A062206 - cp's OEIS frontend

A083282 a(n) = n^(3*n).

1, 64, 19683, 16777216, 30517578125, 101559956668416, 558545864083284007, 4722366482869645213696, 58149737003040059690390169, 1000000000000000000000000000000, 23225154419887808141001767796309131

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Jun 02 2003

If sequence A000312 is used as the domain in the function f(n)=n^3, this sequence would be the resulting range. Also the range of the function f(n)=n^1.5 when sequence A062207 is used as the domain.

G. C. Greubel, Table of n, a(n) for n = 1..150

Cf. A000312, A062206, A062207, A089072.

[n^(3*n): n in [1..30]]; // G. C. Greubel, Nov 01 2022

Table[n^(3*n), {n, 30}] (* G. C. Greubel, Nov 01 2022 *)

[n^(3*n) for n in range(1,31)] # G. C. Greubel, Nov 01 2022

a(n) = A089072(3*n, n). - G. C. Greubel, Nov 01 2022

More terms from Michael Joseph Halm, May 16 2004

A085741 a(n) = T(n)^n, where T() are the triangular numbers (A000217).

Original entry on oeis.org

1, 1, 9, 216, 10000, 759375, 85766121, 13492928512, 2821109907456, 756680642578125, 253295162119140625, 103510234140112521216, 50714860157241037295616, 29345269354638035222576971

Offset: 0

Jon Perry, Jul 21 2003

			a(3) = T(3)^3 = 6^3 = 216.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000217.
Essentially the same as A061718.
Cf. A000312, A062206, A212333.

[((n*(n+1))/2)^n: n in [0..20]]; // Vincenzo Librandi, Sep 14 2011

a:=n->mul(sum(j, j=1..n),k=1..n): seq(a(n), n=0..13); # Zerinvary Lajos, Jun 02 2007

With[{rnn=Range[20]},Join[{1},First[#]^Last[#]&/@Thread[ {Accumulate[ rnn],  rnn}]]] (* Harvey P. Dale, Dec 08 2013 *)

a(n) = (n*(n+1)/2)^n; \\ Michel Marcus, Feb 19 2019

a(n) = ((n*(n+1))/2)^n. - Vincenzo Librandi, Sep 14 2011

More terms from Ray Chandler, Nov 09 2003

A212333 n-th power of the n-th pentagonal number.

Original entry on oeis.org

1, 1, 25, 1728, 234256, 52521875, 17596287801, 8235430000000, 5132188731375616, 4108400332687853397, 4108469075197275390625, 5019255990031848807858176, 7355827511386641000000000000, 12736801848653359358345383963927, 25724477018923486959881583081626689

Offset: 0

Bruno Berselli, May 09 2012

Bruno Berselli, Table of n, a(n) for n = 0..100

Cf. A000312, A085741, A062206.

Magma
```
[(n*(3*n-1)/2)^n: n in [0..14]];
```

Join[{1}, Table[(n ((3 n - 1)/2))^n, {n, 14}]]

a(n) = A000326(n)^n.

A086648 Decimal expansion of the sum n^(-2n) for n=1 through infinity.

Original entry on oeis.org

1, 0, 6, 3, 8, 8, 7, 1, 0, 3, 7, 6, 2, 4, 1, 7, 0, 1, 1, 7, 4, 6, 4, 9, 1, 1, 1, 7, 6, 6, 0, 0, 1, 1, 1, 1, 5, 7, 2, 7, 1, 9, 9, 0, 7, 6, 8, 9, 1, 5, 9, 9, 2, 8, 6, 6, 3, 4, 0, 8, 3, 8, 6, 9, 6, 6, 7, 9, 4, 6, 6, 5, 5, 0, 7, 6, 0, 6, 5, 7, 3, 2, 5, 5, 5, 6, 2, 0, 8, 9, 0, 6, 9, 1, 9, 8, 7, 5, 1, 9, 4, 0, 2, 6, 6

Offset: 1

Christopher M. Tomaszewski (cmt1288(AT)comcast.net), Jul 26 2003

This number is the sum of the inverses of the terms in the sequence A062206.

			1.063887103762417011746491117660011115727199076891599286634083869667...

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A062206, A073009.

suminf(k=1,1./k^(2*k)) \\ Franklin T. Adams-Watters, Mar 23 2010

Clarified definition - R. J. Mathar, Feb 06 2009

More terms from Franklin T. Adams-Watters, Mar 23 2010

A155957 a(n) = (2*n^2)^n.

Original entry on oeis.org

1, 2, 64, 5832, 1048576, 312500000, 139314069504, 86812553324672, 72057594037927936, 76848453272063549952, 102400000000000000000000, 166712830744247830760081408, 325619086145088897570576531456

Offset: 0

Reinhard Zumkeller, Jan 31 2009

nonn

Central terms of the triangle in A155955;

G. C. Greubel, Table of n, a(n) for n = 0..200

Cf. A000079, A062206, A155955.

[(2*n^2)^n: n in [0..30]]; // G. C. Greubel, Sep 14 2018

Table[If[n==0,1,(2*n^2)^n], {n, 0, 30}] (* G. C. Greubel, Sep 14 2018 *)

vector(30, n, n--; (2*n^2)^n) \\ G. C. Greubel, Sep 14 2018

a(n) = A062206(n)*A000079(n).

a(n) = n! * [x^n] 1/(1 + LambertW(-2*n*x)). - Ilya Gutkovskiy, Oct 03 2017

A117812 a(n) = n^(2*n) - 1.

Original entry on oeis.org

0, 0, 15, 728, 65535, 9765624, 2176782335, 678223072848, 281474976710655, 150094635296999120, 99999999999999999999, 81402749386839761113320, 79496847203390844133441535, 91733330193268616658399616008, 123476695691247935826229781856255

Offset: 0

Reinhard Zumkeller, Oct 17 2006

nonn

a(n) = A048861(n)*A014566(n) = A062206(n) - 1.

Cf. A000312, A014566, A048861, A062206.

[n^(2*n) - 1: n in [0..40]]; // Vincenzo Librandi, Dec 26 2010

Join[{0},Table[(n^2)^n-1,{n,17}]] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2011*)

A233203 a(n) = floor(n^n / 2^n).

Original entry on oeis.org

1, 0, 1, 3, 16, 97, 729, 6433, 65536, 756680, 9765625, 139312339, 2176782336, 36972058910, 678223072849, 13363461010158, 281474976710656, 6311342330065435, 150094635296999121, 3773536025353076151, 100000000000000000000, 2785962590401641140642, 81402749386839761113321

Offset: 0

Alex Ratushnyak, Dec 05 2013

nonn

			a(5) = floor(5^5 / 2^5) = floor(3125 / 32) = 97.

Cf. A000079, A000312, A178537 (n^n mod 2^n for odd n), A206344.

Bisection gives: A062206 (even part).

A233203:=n->floor((n/2)^n); seq(A233203(n), n=0..30); # Wesley Ivan Hurt, Feb 26 2014

Table[Floor[(n/2)^n], {n, 0, 30}] (* Wesley Ivan Hurt, Feb 26 2014 *)

a(n) = n^n\2^n; \\ Michel Marcus, Dec 17 2024

for n in range(33): print(n**n >> n, end=', ')

a(n) = floor((n/2)^n).

A290770 a(n) = Product_{k=1..n} k^(2*k).

Original entry on oeis.org

1, 1, 16, 11664, 764411904, 7464960000000000, 16249593066946560000000000, 11020848942410302096869949440000000000, 3102093199396597590886754340698424229232640000000000, 465607547420733489126893933985879279492195953053596584509440000000000

Offset: 0

Ilya Gutkovskiy, Aug 10 2017

G. C. Greubel, Table of n, a(n) for n = 0..27
Eric Weisstein's World of Mathematics, Hyperfactorial
Index entries for sequences related to factorial numbers

Cf. A000178, A001044, A001818, A002109, A051675, A055209, A061742, A062206, A074962, A184877, A185141, A260122.

[1] cat [(&*[k^(2*k): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

Table[Product[k^(2 k), {k, 1, n}], {n, 0, 9}]
Table[Hyperfactorial[n]^2, {n, 0, 9}]
Table[n!^(2 n)/BarnesG[n + 1]^2, {n, 0, 9}]

a(n) = prod(k=1, n, k^(2*k)) \\ Felix Fröhlich, Aug 10 2017

a(n) = A002109(n)^2.
a(n) = A185141(n)/A000178(n-1)^2 for n > 0.
a(n) = (n!)^(2*n)/G(n+1)^2, where G() is the Barnes G-function.
a(n) ~ A^2*exp(-n^2/2)*n^(n*(n+1))*n^(1/6), where A is the Glaisher-Kinkelin constant (A074962).

A350008 a(n) = Sum_{k=0..n} k^(2*k).

Original entry on oeis.org

1, 2, 18, 747, 66283, 9831908, 2186614244, 680409687093, 282155386397749, 150376790683396870, 100150376790683396870, 81502899763630444510191, 79578350103154474577951727, 91812908543371771132977567736

Offset: 0

Seiichi Manyama, Dec 08 2021

nonn

Partial sums of A062206.

Seiichi Manyama, Table of n, a(n) for n = 0..214

Cf. A000330, A062206, A062970, A100262, A249459, A349962.

a[n_] := Sum[If[k == 0, 1, k^(2*k)], {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Dec 08 2021 *)

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

a(n) ~ n^(2*n). - Vaclav Kotesovec, Dec 08 2021

A086815 a(n)=(n-1)n^(2n).

Original entry on oeis.org

0, 16, 1458, 196608, 39062500, 10883911680, 4069338437094, 1970324836974592, 1200757082375992968, 900000000000000000000, 814027493868397611133210, 874465319237299285467856896, 1100799962319223399900795392108

Offset: 1

Benoit Cloitre, Aug 06 2003

nonn

(-1)*determinant of the 2n X 2n matrix M_(i,j)=j if (i+j) is multiple of n, M_(i,j)=i otherwise.

			For n=3 the matrix is : [1 2 1 1 5 1], [1 2 2 4 2 2], [3 3 3 3 3 6], [4 2 4 4 5 4], [1 5 5 4 5 5], [6 6 3 6 6 6]

Cf. A062206.

Table[(n-1)n^(2n),{n,20}] (* Harvey P. Dale, Jul 23 2013 *)

PARI
```
a(n)=(n-1)*n^(2*n)
```

cp's OEIS Frontend

A083282 a(n) = n^(3*n).

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A085741 a(n) = T(n)^n, where T() are the triangular numbers (A000217).

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A212333 n-th power of the n-th pentagonal number.

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Magma

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Formula

A086648 Decimal expansion of the sum n^(-2n) for n=1 through infinity.

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PARI

Extensions

A155957 a(n) = (2*n^2)^n.

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Magma

Mathematica

PARI

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A117812 a(n) = n^(2*n) - 1.

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Magma

Mathematica

A233203 a(n) = floor(n^n / 2^n).

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A086815 a(n)=(n-1)n^(2n).