A004193 -id:A004193 - cp's OEIS frontend

A097749 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. Let A(n,k) be the triangle in A097474. Then T(n,k) is defined by the orthogonality relations Sum_{j=i..r} T(r,j)A(j,i)2^-floor((j+3)/2) = 0 if i != r, = (2r+1)!/(r!*2^r) if i = r.

2, 1, 2, -1, 10, 6, 5, -35, 105, 30, -63, 420, -882, 1260, 210, 1576, -10395, 20790, -20790, 17325, 1890, -68409, 450450, -891891, 849420, -495495, 270270, 20790, 4729726, -31126095, 61486425, -57972915, 32207175, -12297285, 4729725, 270270

Offset: 0

N. J. A. Sloane, Sep 21 2004

			Triangle begins:
2
1 2
-1 10 6
5 -35 105 30
-63 420 -882 1260 210

H. W. Gould, Power sum identities for arbitrary symmetric arrays, SIAM J. Appl. Math., 17 (1969), 307-316.

Cf. A097474, A097801. Row sums give A001147. Is the left-hand edge A004193?

More terms from Sean A. Irvine, Mar 25 2013

A238087 A129779(n+2)/A006954(n).

Original entry on oeis.org

-1, 1, -1, 1, -5, 63, -315, 99, -675675, 135135, -1640925, 83329155, -4583103525, 5791834125, -71152682225625, 14230536445125, -26797763435625, 24833960277501375, -73881031825566590625, 8546099690638125, -106610328924292590271875

Offset: 0

Paul Curtz, Feb 24 2014

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In A129779 or A097801 there is no mention of a link with the Bernoulli numbers.

It appears that a(n+6) is divisible by 4*(n+4)^2-1 and that the sum of the digits of a(n+5) is a multiple of 9.

			a(0) = -1/1, a(1) = 2/2, a(2) = -6/6, a(3) = 30/30, a(4) = -210/42 =-5.

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

Cf. A000466.

a[0] = -1; a[1] = 1; a[n_] := (-1)^(n+1)*(2*n)!/(2^(n-1)*n!*Denominator[BernoulliB[2*n-2]]); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 24 2014 *)

a(n+2) = -A004193(n+1)/A000367(n+1).

More terms from Jean-François Alcover, Feb 24 2014

A263445 a(n) = (2n+1)(n+1)!Bernoulli(2n).

Original entry on oeis.org

1, 1, -1, 4, -36, 600, -16584, 705600, -43751232, 3790108800, -443539877760, 68218849036800, -13478425925184000, 3355402067989171200, -1035218714714606822400, 390189256983139461120000, -177430554756972746695065600, 96269372301568677170319360000

Offset: 0

Vladimir Reshetnikov, Oct 18 2015

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Robert Israel, Table of n, a(n) for n = 0..210
Index entries for sequences related to Bernoulli numbers.

Bernoulli numbers are A000367/A002445. Cf. A004193, A001332, A000182, A001469.

seq((2*n+1)*(n+1)!*bernoulli(2*n), n=0..50); # Robert Israel, Oct 18 2015

Table[(2n + 1) (n + 1)! BernoulliB[2n], {n, 0, 17}]

vector(30, n, n--; (2*n+1)*(n+1)!*bernfrac(2*n)) \\ Altug Alkan, Oct 18 2015

from math import factorial
from sympy import bernoulli
def A263445(n): return (2*n+1)*factorial(n+1)*bernoulli(2*n) # Chai Wah Wu, May 18 2022

a(n) = (2n+1)*(n+1)!*Bernoulli(2n).

a(n) ~ (-1)^(n+1)*8*sqrt(2)*n^3*(n/e)^(3*n)*Pi^(1-2*n). - Vladimir Reshetnikov, Sep 05 2016

A168136 a(n) = Bernoulli(2n)*(2n+1)!/n!.

Original entry on oeis.org

1, 1, -2, 20, -504, 25200, -2189088, 302702400, -62564261760, 18427508985600, -7449695786856960, 4010313259477324800, -2803674333549374208000, 2492728196309155284480000, -2768630339381333070099456000

Offset: 0

Franklin T. Adams-Watters, Nov 19 2009

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Given by alternating sums of coefficients of polynomials spawned by sums of binomial(n,k)^2*k^(2m). - John M. Campbell, Nov 30 2011

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

Cf. A001332, A004193.

[Bernoulli(2*n)*Factorial(2*n+1)/Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 14 2016

Table[BernoulliB[2*n]*(2*n+1)!/n!, {n, 0, 20}]

PARI
```
a(n)=bernfrac(2*n)*(2*n+1)!/n!
```

A335750 a(n) = numerator(Bernoulli(2n)(1/2 - n)! / sqrt(Pi)).

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 11056, 32, 231488, 5614976, 44700416, 39773696, 242036829184, 1347442688, 13896827482112, 14116194346606592, 126309515939299328, 4968569161351168, 1724597636500912693116928, 20212640119738990592, 68441268157533158650937344, 796968953534517505001259008

Offset: 0

Peter Luschny, Jun 20 2020

			r(n) = 1/2, 1/6, 1/15, 2/63, 4/225, 8/693, 11056/1289925, 32/4455, ...

Cf. A335751 (denominator), A000367/A002445, A004193.

a := n -> bernoulli(2*n)*(1/2 - n)! / sqrt(Pi):
seq(numer(simplify(a(n))), n = 0..21);

a(n) = numerator(-2*n*Zeta(1 - 2*n)*(1/2 - n)! / sqrt(Pi)) for n >= 1.

A335751 a(n) = denominator(Bernoulli(2n)(1/2 - n)! / sqrt(Pi)).

Original entry on oeis.org

2, 6, 15, 63, 225, 693, 1289925, 4455, 34459425, 808782975, 5685805125, 4106936925, 18767808934875, 72977109975, 491292329653125, 305714614450620375, 1578522255175490625, 33864491287501875, 6076788748684677645496875, 34996278233163121875, 55478375013295336399171875

Offset: 0

Peter Luschny, Jun 20 2020

			r(n) = 1/2, 1/6, 1/15, 2/63, 4/225, 8/693, 11056/1289925, 32/4455, ...

Cf. A335750 (numerator), A000367/A002445, A004193.

a := n -> bernoulli(2*n)*(1/2 - n)! / sqrt(Pi):
seq(denom(simplify(a(n))), n = 0..21);

a(n) = denominator(-2*n*Zeta(1 - 2*n)*(1/2 - n)! / sqrt(Pi)) for n >= 1.

cp's OEIS Frontend

A097749 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. Let A(n,k) be the triangle in A097474. Then T(n,k) is defined by the orthogonality relations Sum_{j=i..r} T(r,j)*A(j,i)*2^-floor((j+3)/2) = 0 if i != r, = (2r+1)!/(r!*2^r) if i = r.

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A238087 A129779(n+2)/A006954(n).

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Mathematica

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A263445 a(n) = (2n+1)*(n+1)!*Bernoulli(2n).

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A168136 a(n) = Bernoulli(2n)*(2n+1)!/n!.

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Magma

Mathematica

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A335750 a(n) = numerator(Bernoulli(2*n)*(1/2 - n)! / sqrt(Pi)).

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Author

Keywords

Examples

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Programs

Maple

Formula

A335751 a(n) = denominator(Bernoulli(2*n)*(1/2 - n)! / sqrt(Pi)).

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Formula

A097749 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. Let A(n,k) be the triangle in A097474. Then T(n,k) is defined by the orthogonality relations Sum_{j=i..r} T(r,j)A(j,i)2^-floor((j+3)/2) = 0 if i != r, = (2r+1)!/(r!*2^r) if i = r.

A263445 a(n) = (2n+1)(n+1)!Bernoulli(2n).

A335750 a(n) = numerator(Bernoulli(2n)(1/2 - n)! / sqrt(Pi)).

A335751 a(n) = denominator(Bernoulli(2n)(1/2 - n)! / sqrt(Pi)).