A085527 -id:A085527 - cp's OEIS frontend

A274265 a(n) = (3*n - 1)^(n-1).

1, 5, 64, 1331, 38416, 1419857, 64000000, 3404825447, 208827064576, 14507145975869, 1125899906842624, 96549157373046875, 9065737908494995456, 925103102315013629321, 101938319743841411792896, 12063348350820368238715343, 1525878906250000000000000000

Offset: 1

Peter Bala, Jun 19 2016

Compare with A052752.

Cf. A000169, A052752, A085527, A274266, A274267, A274269.

[(3*n-1)^(n-1): n in [1..20]]; // Vincenzo Librandi, Jun 20 2016

A274265 := n -> (3*n - 1)^(n-1):
seq(A274265(n), n = 1..20);

Table[(3*n-1)^(n-1), {n,1,25}] (* G. C. Greubel, Jun 19 2016 *)

E.g.f. A(x) = 1 - exp(-1/3*T(3*x)) = x + 5*x^2/2! + 8^2*x^3/3! + 11^3*x^4/4! + 14^4*x^5/5! + ..., where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
A(x) = series reversion( (1 - x)^3*log(1/(1 - x)) ). See A274266.
1 - A(x) = exp(-x/(1 - A(x))^3) = exp(-x/(exp(-3*x/(exp(-3*x/ ...))))).
1 - A(-x*exp(3*x)) = exp(x) = 1/(1 - A(x*exp(-3*x))).
1/(1 - A(x)) = Sum_{n >= 0} (3*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052752.

A085528 a(n) = (2*n+1)^(n+1).

Original entry on oeis.org

1, 9, 125, 2401, 59049, 1771561, 62748517, 2562890625, 118587876497, 6131066257801, 350277500542221, 21914624432020321, 1490116119384765625, 109418989131512359209, 8629188747598184440949, 727423121747185263828481, 65273511648264442971824673

Offset: 0

N. J. A. Sloane, Jul 05 2003

a(n) is the number of polynomials of degree at most n with integer coefficients all having absolute value <= n.

a(n-1) is the number of nodes in the canonical automaton for the affine Weyl group of type D_n. - Tom Edgar, May 12 2016

Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups. Graduate Texts in Mathematics, 231. Springer, New York, 2005.

Vincenzo Librandi, Table of n, a(n) for n = 0..100

Cf. A000169, A085527, A099753, A214406, A253299.

List([0..20], n-> (2*n+1)^(n+1)); # G. C. Greubel, Sep 03 2019

[(2*n+1)^(n+1): n in [0..20]]; // Vincenzo Librandi, May 04 2011

seq((2*n+1)^(n+1), n=0..20); # G. C. Greubel, Sep 03 2019

Table[(2*n+1)^(n+1), {n,0,20}] (* Vladimir Joseph Stephan Orlovsky, Sep 05 2009, modified by G. C. Greubel, Sep 03 2019 *)

vector(20, n, (2*n-1)^n) \\ G. C. Greubel, Sep 03 2019

[(2*n+1)^(n+1) for n in (0..20)] # G. C. Greubel, Sep 03 2019

From Peter Bala, Aug 06 2012: (Start)
E.g.f.: d/dx{(2*x/T(2*x))^(1/2)*1/(1 - T(2*x))} = 1 + 9*x + 125*x^2/2! + ..., where T(x) is the tree function sum {n >= 1} n^(n-1)*x^n/n! of A000169.
For r = 0, 1, 2, ... the e.g.f. for the sequence (2*n+1)^(n+r) can be expressed in terms of the function U(z) = sum {n >= 0} (2*n+1)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 1, and the resulting e.g.f. is 1/z*U(z)*(1 + U(z)^2 )/(1 - U(z)^2)^3 taken at z = sqrt(2*x).
(End)
Sum_{n>=0} (-1)^n/a(n) = A253299. - Amiram Eldar, Jun 25 2021

A088961 Zigzag matrices listed entry by entry.

Original entry on oeis.org

3, 5, 5, 5, 10, 14, 14, 7, 14, 21, 21, 7, 21, 35, 42, 48, 27, 9, 48, 69, 57, 36, 27, 57, 78, 84, 9, 36, 84, 126, 132, 165, 110, 44, 11, 165, 242, 209, 121, 55, 110, 209, 253, 220, 165, 44, 121, 220, 297, 330, 11, 55, 165, 330, 462

Offset: 1

Paul Boddington, Oct 28 2003

For each n >= 1 the n X n matrix Z(n) is constructed as follows. The i-th row of Z(n) is obtained by generating a hexagonal array of numbers with 2*n+1 rows, 2*n numbers in the odd numbered rows and 2*n+1 numbers in the even numbered rows. The first row is all 0's except for two 1's in the i-th and the (2*n+1-i)th positions. The remaining rows are generated using the same rule for generating Pascal's triangle. The i-th row of Z(n) then consists of the first n numbers in the bottom row of our array.
For example the top row of Z(2) is [5,5], found from the array:
. 1 0 0 1
1 1 0 1 1
. 2 1 1 2
2 3 2 3 2
. 5 5 5 5
Zigzag matrices have remarkable properties. Here is a selection:
1) Z(n) is symmetric.
2) det(Z(n)) = A085527(n).
3) tr(Z(n)) = A033876(n-1).
4) If 2*n+1 is a power of a prime p then all entries of Z(n) are multiples of p.
5) If 4*n+1 is a power of a prime p then the dot product of any two distinct rows of Z(n) is a multiple of p.
6) It is always possible to move from the bottom left entry of Z(n) to the top right entry using only rightward and upward moves and visiting only odd numbers.
A001700(n) = last term of last row of Z(n): a(A000330(n-1)) = A001700(n); A230585(n) = first term of first row of Z(n): a(A056520(n-1)) = A230585(n); A051417(n) = greatest common divisor of entries of Z(n). - Reinhard Zumkeller, Oct 25 2013

			The first five values are 3, 5, 5, 5, 10 because the first two zigzag matrices are [[3]] and [[5,5],[5,10]].

Reinhard Zumkeller, Matrices Z(n): n = 1..30, flattened

Cf. A085527, A003876.

a088961 n = a088961_list !! (n-1)
a088961_list = concat $ concat $ map f [1..] where
   f x = take x $ g (take x (1 : [0,0..])) where
     g us = (take x $ g' us) : g (0 : init us)
     g' vs = last $ take (2 * x + 1) $
                    map snd $ iterate h (0, vs ++ reverse vs)
   h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
-- Reinhard Zumkeller, Oct 25 2013

Flatten[Table[Binomial[2n,n+j-i]-Binomial[2n,n+i+j]+ Binomial[2n, 3n+1-i-j], {n,5},{i,n},{j,n}]] (* Harvey P. Dale, Dec 15 2011 *)

The ij entry of Z(n) is binomial(2*n, n+j-i) - binomial(2*n, n+i+j) + binomial(2*n, 3*n+1-i-j).

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)

A099753 a(n) = (2*n+1)^(n+2).

Original entry on oeis.org

1, 27, 625, 16807, 531441, 19487171, 815730721, 38443359375, 2015993900449, 116490258898219, 7355827511386641, 504036361936467383, 37252902984619140625, 2954312706550833698643, 250246473680347348787521, 22550116774162743178682911, 2154025884392726618070214209

Offset: 0

Kari Lajunen (Kari.Lajunen(AT)Welho.com), Nov 11 2004

G. C. Greubel, Table of n, a(n) for n = 0..345

Cf. A000169, A085527, A085528, A214406.

List([0..30], n-> (2*n+1)^(n+2)); # G. C. Greubel, Sep 03 2019

[(2*n+1)^(n+2): n in [0..30]]; // G. C. Greubel, Sep 03 2019

seq((2*n+1)^(n+2), n=0..30); # G. C. Greubel, Sep 03 2019

Table[(2*n+1)^(n+2), {n,0,30}] (* G. C. Greubel, Sep 03 2019 *)

vector(30, n, (2*n-1)^(n+1)) \\ G. C. Greubel, Sep 03 2019

[(2*n+1)^(n+2) for n in (0..30)] # G. C. Greubel, Sep 03 2019

From Peter Bala, Aug 06 2012: (Start)
E.g.f.: d^2/dx^2{(2*x/T(2*x))^(3/2)*1/(1 - T(2*x))} = 1 + 27*x + 625*x^2/2! + ..., where T(x) is the tree function sum {n >=1} n^(n-1)*x^n/n! of A000169.
For r = 0, 1, 2, ..., the e.g.f. for the sequence (2*n+1)^(n+r) can be expressed in terms of the function U(z) = sum {n >= 0} (2*n+1)^(n-1)*z^(2*n+1)/(2^n*n!). See A214406 for details. In the present case, r = 2, and the resulting e.g.f. is 1/z*U(z)*(1 + 8*U(z)^2 + 3*U(z)^4)/(1 - U(z)^2)^5 taken at z = sqrt(2*x).
(End)

Terms a(13) onward added by G. C. Greubel, Sep 03 2019

A085529 a(n) = (2n+1)^(2n+1).

Original entry on oeis.org

1, 27, 3125, 823543, 387420489, 285311670611, 302875106592253, 437893890380859375, 827240261886336764177, 1978419655660313589123979, 5842587018385982521381124421, 20880467999847912034355032910567, 88817841970012523233890533447265625, 443426488243037769948249630619149892803

Offset: 0

N. J. A. Sloane, Jul 05 2003

nonn

a(n) == 2*n + 1 (mod 24). - Mathew Englander, Aug 16 2020

Cf. A000312, A005408, A016754, A085527, A085528, A085530, A085531, A085532, A085533, A085534, A085535.

Cf. A073009, A083648.

#^#&/@Range[1,31,2] (* Harvey P. Dale, Oct 05 2019 *)

From Mathew Englander, Aug 16 2020: (Start)
a(n) = A000312(2*n + 1).
a(n) = A016754(n)^n * (2*n + 1).
a(n) = A085527(n)^2 * (2*n + 1).
a(n) = A085528(n)^2 / (2*n + 1).
a(n) = A085530(n) * A005408(n).
a(n) = A085531(n) * A016754(n).
a(n) = A085532(n)^2 - A215265(2*n + 1).
a(n) = A085533(n) + A045531(2*n + 1).
a(n) = A085534(n+1) - A007781(2*n + 1).
a(n) = A085535(n+1) - A055869(2*n + 1).
(End)
Sum_{n>=0} 1/a(n) = (A073009 + A083648)/2 = 1.0373582538... . - Amiram Eldar, May 17 2022

A363398 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (2*j + 1)^n), (secant case).

Original entry on oeis.org

1, 3, 3, 7, 36, 25, 15, 297, 625, 343, 31, 2106, 10000, 14406, 6561, 63, 13851, 131250, 369754, 413343, 161051, 127, 87480, 1546875, 7529536, 15411789, 14172488, 4826809, 255, 540189, 17109375, 134237509, 444816117, 721025327, 564736653, 170859375

Offset: 0

Peter Luschny, May 31 2023

Here we give an inclusion-exclusion representation of 2^n*Euler(n) (see A122045 and A002436), in A363399 we give such a representation for 2^n*Euler(n, 1) = A155585(n), and in A363400 one for the combined sequences.

			The triangle T(n, k) starts:
  [0]   1;
  [1]   3,      3;
  [2]   7,     36,       25;
  [3]  15,    297,      625,       343;
  [4]  31,   2106,    10000,     14406,      6561;
  [5]  63,  13851,   131250,    369754,    413343,    161051;
  [6] 127,  87480,  1546875,   7529536,  15411789,  14172488,   4826809;
  [7] 255, 540189, 17109375, 134237509, 444816117, 721025327, 564736653, 170859375;

Index entries for sequences related to Euler numbers.

Cf. A122045 (alternating row sums), A363396 (row sums), A126646 (column 0), A085527 (main diagonal), A141475 (central terms).
Cf. A363399 (tangent case), A363400 (combined case).
Cf. A122045, A002436.

P := (n, x) -> add(add(x^j*binomial(k, j)*(2*j + 1)^n, j=0..k)*2^(n-k), k=0..n):
T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..7);

(* From Detlef Meya, Oct 04 2023: (Start) *)
T[n_, k_] := (2*k+1)^n*(2^(n+1) - Sum[Binomial[n+1, j], {j,0,k}]);
(* Or: *)
T[n_, k_] := (2*k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)

Sum_{k=0..n} (-1)^k*T(n, k) = 2^n*Euler(n) = 4^n*Euler(n, 1/2).
(Sum_{k=0..n} (-1)^k*T(n, k)) / 2^n = Euler(n) = 2^n*Euler(n, 1/2) = A122045(n).
Sum_{k=0..2*n} (-1)^k*T(2*n, k) = 4^n*Euler(2*n) = 16^n*Euler(2*n, 1/2) = (-1)^n*A002436(n).
From Detlef Meya, Oct 04 2023: (Start)
T(n, k) = (2*k + 1)^n * binomial(n+1, k+1) * hypergeom([1, k-n], [k+2], -1).
T(n, k) = (2*k + 1)^n * (2^(n + 1) - Sum_{j=0..k} binomial(n+1, j)). (End)

A376037 E.g.f. satisfies A(x) = (exp(x / (1 - A(x))^2) - 1) / (1 - A(x)).

Original entry on oeis.org

0, 1, 7, 115, 3047, 111771, 5244555, 299941195, 20239069807, 1574068019851, 138641219870243, 13640672949173403, 1482772864485867399, 176478769995088245595, 22825571074271407363771, 3187825736999237502879019, 478120273969744650293424095

Offset: 0

Seiichi Manyama, Sep 07 2024

nonn

Index entries for reversions of series

Cf. A371329, A376036.

Cf. A085527, A371371.

a(n) = sum(k=1, n, (2*n+2*k-2)!/(2*n+k-1)!*stirling(n, k, 2));

a(n) = Sum_{k=1..n} (2*n+2*k-2)!/(2*n+k-1)! * Stirling2(n,k).

E.g.f.: Series_Reversion( (1 - x)^2 * log(1 + x * (1 - x)) ).

A333420 Table T(n,k) read by upward antidiagonals. T(n,k) is the maximum value of Product_{i=1..n} Sum_{j=1..k} r[(i-1)*k+j] among all permutations r of {1..kn}.

Original entry on oeis.org

1, 2, 3, 6, 25, 6, 24, 343, 110, 10, 120, 6561, 3375, 324, 15, 720, 161051, 144400, 17576, 756, 21, 5040, 4826809, 7962624, 1336336, 64000, 1521, 28, 40320, 170859375, 535387328, 130691232, 7595536, 185193, 2756, 36, 3628800, 6975757441

Offset: 1

Chai Wah Wu, Mar 23 2020

A dual sequence to A331889.
   k         1          2         3         4       5        6         7       8       9
  --------------------------------------------------------------------------------------
n  1|        1          3         6        10      15       21        28      36      45
   2|        2         25       110       324     756     1521      2756    4624    7310
   3|        6        343      3375     17576   64000   185193    456533 1000000 2000376
   4|       24       6561    144400   1336336 7595536 31640625 106131204
   5|      120     161051   7962624 130691232
   6|      720    4826809 535387328
   7|     5040  170859375
   8|    40320 6975757441
   9|  3628800
  10| 39916800

Chai Wah Wu, On rearrangement inequalities for multiple sequences, arXiv:2002.10514 [math.CO], 2020.

Cf. A000142, A000217, A085527, A331889.

from itertools import combinations, permutations
from sympy import factorial
def T(n,k): # T(n,k) for A333420
    if k == 1:
        return int(factorial(n))
    if n == 1:
        return k*(k+1)//2
    if k % 2 == 0 or (k >= n-1 and n % 2 == 1):
        return (k*(k*n+1)//2)**n
    if k >= n-1 and n % 2 == 0 and k % 2 == 1:
        return ((k**2*(k*n+1)**2-1)//4)**(n//2)
    nk = n*k
    nktuple = tuple(range(1,nk+1))
    nkset = set(nktuple)
    count = 0
    for firsttuple in combinations(nktuple,n):
        nexttupleset = nkset-set(firsttuple)
        for s in permutations(sorted(nexttupleset),nk-2*n):
            llist = sorted(nexttupleset-set(s),reverse=True)
            t = list(firsttuple)
            for i in range(0,k-2):
                itn = i*n
                for j in range(n):
                        t[j] += s[itn+j]
            t.sort()
            w = 1
            for i in range(n):
                w *= llist[i]+t[i]
            if w > count:
                count = w
    return count

T(n,k) <= floor((k*(k*n+1)/2)^n) with equality if k = 2*t+n*u for nonnegative integers t and u.
T(n,1) = n! = A000142(n).
T(1,k) = k*(k+1)/2 = A000217(k).
T(n,2) = (2*n+1)^n = A085527(n).
If n is even, k is odd and k >= n-1, then T(n,k) = ((k^2*(k*n+1)^2-1)/4)^(n/2).

A229212 Square array of numerators of t(n,k) = (1+1/(k*n))^n, read by descending antidiagonals.

Original entry on oeis.org

2, 3, 9, 4, 25, 64, 5, 49, 343, 625, 6, 81, 1000, 6561, 7776, 7, 121, 2197, 28561, 161051, 117649, 8, 169, 4096, 83521, 1048576, 4826809, 2097152, 9, 225, 6859, 194481, 4084101, 47045881, 170859375, 43046721, 10, 289

Offset: 1

Jean-François Alcover, Sep 16 2013

Limit(t(n,k), n -> infinity) = exp(1/k).
1st row = A020725
2nd row = A016754
3rd row = A016779
4th row = A016816
5th row = A016865
1st column = A000169
2nd column = A085527

			Table of fractions begins:
   2,       3/2,        4/3,         5/4, ...
  9/4,     25/16,      49/36,       81/64, ...
64/27,   343/216,   1000/729,    2197/1728, ...
625/256, 6561/4096, 28561/20736, 83521/65536, ...
...
Table of numerators begins:
2,      3,     4,     5, ...
9,     25,    49,    81, ...
64,   343,  1000,  2197, ...
625, 6561, 28561, 83521, ...
...
Triangle of antidiagonals begins:
2;
3, 9;
4, 25, 64;
5, 49, 343, 625;
...

Cf. A229213(denominators), A016754, A016779, A016816, A016865, A000169, A085527.

t[n_, k_] := (1+1/(k*n))^n; Table[t[n-k+1, k], {n, 1, 9}, {k, n, 1, -1}] // Flatten // Numerator

cp's OEIS Frontend

A274265 a(n) = (3*n - 1)^(n-1).

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A085528 a(n) = (2*n+1)^(n+1).

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A088961 Zigzag matrices listed entry by entry.

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

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A099753 a(n) = (2*n+1)^(n+2).

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A085529 a(n) = (2n+1)^(2n+1).

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A363398 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (2*j + 1)^n), (secant case).

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