A085529 -id:A085529 - cp's OEIS frontend

A085534 a(n) = (2n)^(2n).

1, 4, 256, 46656, 16777216, 10000000000, 8916100448256, 11112006825558016, 18446744073709551616, 39346408075296537575424, 104857600000000000000000000, 341427877364219557396646723584, 1333735776850284124449081472843776, 6156119580207157310796674288400203776

Offset: 0

N. J. A. Sloane, Jul 05 2003

nonn

All terms are both perfect squares and numbers of the form n^n. - William Boyles, Jul 31 2015
Intersection of A000290 and A000312. - Michel Marcus, Aug 04 2015
Intersection of A005843 and A000312. - Robert Israel, Aug 04 2015
The number of sequences of length 2n using 2n symbols. - Washington Bomfim, Jan 14 2020

Alois P. Heinz, Table of n, a(n) for n = 0..100

Cf. A000312, A073009, A083648, A085529.

Column k=0 of A246070.

{1}~Join~Table[(2 n)^(2 n), {n, 1, 4!}] (* Michael De Vlieger, Aug 04 2015 *)

def A085534(n): return (m:=n<<1)**m # Chai Wah Wu, Nov 18 2022

a(n) = A000312(2*n). - Michel Marcus, Jul 31 2015
a(n) = A062971(n)^2. - Michel Marcus, Aug 04 2015
a(n) = [x^(2*n)] 1/(1 - 2*n*x). - Ilya Gutkovskiy, Oct 10 2017
Sum_{n>=0} 1/a(n) = 1 + (A073009-A083648)/2 = 1.2539277431... . - Amiram Eldar, May 17 2022

A085530 a(n) = (2n+1)^(2n).

Original entry on oeis.org

1, 9, 625, 117649, 43046721, 25937424601, 23298085122481, 29192926025390625, 48661191875666868481, 104127350297911241532841, 278218429446951548637196401, 907846434775996175406740561329, 3552713678800500929355621337890625, 16423203268260658146231467800709255289

Offset: 0

N. J. A. Sloane, Jul 05 2003

a(n)/4^n is the square of the determinant of a (2*n+1) X (2*n+1) matrix with elements M(j,k) = cos(Pi*j*k/n). See the MathOverflow link. - Hugo Pfoertner, Sep 18 2021

Vincenzo Librandi, Table of n, a(n) for n = 0..190
Zhi-Wei Sun, Fedor Petrov, A surprising identity, discussion in MathOverflow, Jan 17 2019.

Cf. A005408, A005843, A016754, A085527, A085529.

[(2*n+1)^(2*n): n in [0..13]]; // Vincenzo Librandi, Feb 26 2013

Table[(2 n + 1)^(2 n), {n, 0, 20}] (* Vincenzo Librandi, Feb 26 2013 *)

From Mathew Englander, Aug 14 2020: (Start)
a(n) = A085527(n)^2.
a(n) = A085529(n)/(2*n + 1).
(End)
From Alois P. Heinz, Aug 14 2020: (Start)
a(n) = A016754(n)^n.
a(n) = A005408(n)^A005843(n). (End)

A061787 a(n) = Sum_{k=1..n} (2k-1)^(2k-1).

Original entry on oeis.org

1, 28, 3153, 826696, 388247185, 285699917796, 303160806510049, 438197051187369424, 827678458937524133601, 1979247334119251113257580, 5844566265720101772494382001, 20886312566113632136127527292568

Offset: 1

Amarnath Murthy, May 25 2001

			a(3) = 1 + 3^3 + 5^5 = 3153.

Harry J. Smith, Table of n, a(n) for n = 1..100

Partial sums of A085529.

List([1..13],n->Sum([1..n],k->(2*k-1)^(2*k-1))); # Muniru A Asiru, Jul 01 2018

Array[Sum[(2 k - 1)^(2 k - 1), {k, #}] &, 12] (* Michael De Vlieger, Jul 01 2018 *)
Accumulate[#^#&/@Range[1,25,2]] (* Harvey P. Dale, Jul 16 2022 *)

{ a=n=0; forstep (p=1, 200, 2, write("b061787.txt", n++, " ", a+=p^p) ) } \\ Harry J. Smith, Jul 28 2009

a(n) ~ exp(-1) * 2^(2*n-1) * n^(2*n-1). - Vaclav Kotesovec, Dec 09 2021

Corrected and extended by Jason Earls, May 27 2001

cp's OEIS Frontend

A085534 a(n) = (2n)^(2n).

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A085530 a(n) = (2n+1)^(2n).

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A061787 a(n) = Sum_{k=1..n} (2k-1)^(2k-1).

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