A192366 -id:A192366 - cp's OEIS frontend

A117575 Expansion of (1-x^3)/((1-x)(1+2x^2)).

1, 1, -1, -2, 2, 4, -4, -8, 8, 16, -16, -32, 32, 64, -64, -128, 128, 256, -256, -512, 512, 1024, -1024, -2048, 2048, 4096, -4096, -8192, 8192, 16384, -16384, -32768, 32768, 65536, -65536, -131072, 131072, 262144, -262144, -524288, 524288

Offset: 0

Paul Barry, Mar 29 2006

Row sums of A116949.
From Paul Curtz, Oct 24 2012: (Start)
b(n) = abs(a(n)) = A158780(n+1) = 1,1,1,2,2,4,4,8,8,8,... .
Consider the autosequence (that is a sequence whose inverse binomial transform is equal to the signed sequence) of the first kind of the example. Its numerator is A046978(n), its denominator is b(n). The numerator of the first column is A075553(n).
The denominator corresponding to the 0's is a choice.
The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)

			   0/1,  1/1    1/1,   1/2,   0/2,  -1/4,  -1/4,  -1/8, ...
   1/1,  0/1,  -1/2,  -1/2,  -1/4,   0/4,   1/8,   1/8, ...
  -1/1, -1/2,   0/2,   1/4,   1/4,   1/8,   0/8, -1/16, ...
   1/2,  1/2,   1/4,   0/4   -1/8,  -1/8, -1/16,  0/16, ...
   0/2, -1/4,  -1/4,  -1/8,   0/8,  1/16,  1/16,  1/32, ...
  -1/4,  0/4,   1/8,   1/8,  1/16,  0/16, -1/32, -1/32, ...
   1/4,  1/8,   0/8, -1/16, -1/16, -1/32,  0/32,  1/64, ...
  -1/8, -1/8, -1/16,  0/16,  1/32,  1/32,  1/64,  0/64. - _Paul Curtz_, Oct 24 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,-2).

Cf. A016116, A046978, A075553, A116949, A122016, A191754/A192366.

The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - N. J. A. Sloane, Jul 14 2022

[1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // G. C. Greubel, Apr 19 2023

CoefficientList[Series[(1-x^3)/((1-x)(1+2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,-2},{1,1,-1},45] (* Harvey P. Dale, Apr 09 2018 *)

a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ Charles R Greathouse IV, Jan 31 2012

def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
[A117575(n) for n in range(51)] # G. C. Greubel, Apr 19 2023

a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) for n >= 3.
a(n) = (cos(Pi*n/2) + sin(Pi*n/2)) * (2^((n-1)/2)*(1-(-1)^n)/2 + 2^((n-2)/2)*(1+(-1)^n)/2 + 0^n/2).
a(n+1) = Sum_{k=0..n} A122016(n,k)*(-1)^k. - Philippe Deléham, Jan 31 2012
E.g.f.: (1 + cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x))/2. - Stefano Spezia, Feb 05 2023
a(n) = (-1)^floor(n/2)*2^floor((n-1)/2), with a(0) = 1. - G. C. Greubel, Apr 19 2023

A191754 Numerators of a companion to the Bernoulli numbers.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 41, 41, -589, -589, 8317, 8317, -869807, -869807, 43056421, 43056421, -250158593, -250158593, 67632514765, 67632514765, -1581439548217, -1581439548217

Offset: 0

Paul Curtz, Jun 15 2011

The companion to the Bernoulli numbers BC(0, m) = A191754(m)/A192366(m) is, just like the Bernoulli numbers T(0, m) = A164555(m)/A027642(m), see A190339 for the T(n, m), an autosequence of the second kind, i.e., its inverse binomial transform is the sequence signed.
In order to construct the companion array BC(n, m) we use the following rules: the main diagonal BC(n, n) = 0, the first upper diagonal BC(n, n+1) = T(n, n+1) and recurrence relation BC(n, m) = BC(n-1, m+1) - BC(n-1, m). The companion to the Bernoulli numbers appears in the first row of the BC(n, m) array, i.e., BC(0, m) = A191754(m)/A192366(m).
For the denominators of the companion to the Bernoulli numbers see A192366.

			The first few rows of the BC(n,m) matrix are:
0,        1/2,   1/2,    1/3,    1/6,    1/15,    1/30,
1/2,        0,  -1/6,   -1/6,  -1/10,   -1/30,  -1/210,
-1/2,    -1/6,     0,   1/15,   1/15,    1/35,  -1/105,
1/3,      1/6,  1/15,      0, -4/105,  -4/105,       0,
-1/6,   -1/10, -1/15, -4/105,      0,   4/105,   4/105,
1/15,    1/30,  1/35,  4/105,  4/105,       0, -16/231,
-1/30, -1/210, 1/105,      0, -4/105, -16/231,       0,

Cf. A000012, A027642, A164555, A190339, A191754, A192366.

nmax:=26: mmax:=nmax: A164555:=proc(n): if n=1 then 1 else numer(bernoulli(n)) fi: end: A027642:=proc(n): if n=1 then 2 else denom(bernoulli(n)) fi: end: for m from 0 to 2*mmax do T(0,m):=A164555(m)/A027642(m) od: for n from 1 to nmax do for m from 0 to 2*mmax do T(n,m):= T(n-1,m+1)-T(n-1,m) od: od: for n from 0 to nmax do BC(n,n):=0: BC(n,n+1) := T(n,n+1) od: for m from 2 to 2*mmax do for n from 0 to m-2 do BC(n,m):=BC(n,m-1) + BC(n+1,m-1) od: od: for n from 0 to 2*nmax do BC(n,0):=(-1)^(n+1)*BC(0,n) od: for m from 1 to mmax do for n from 2 to 2*nmax do BC(n,m) := BC(n,m-1) + BC(n+1,m-1) od: od: for n from 0 to nmax do seq(BC(n,m),m=0..mmax) od: seq(BC(0,n),n=0..nmax): seq(numer(BC(0,n)),n=0..nmax); # Johannes W. Meijer, Jul 02 2011

max = 26; b[n_] := BernoulliB[n]; b[1]=1/2; bb = Table[b[n], {n, 0, max}]; diff = Table[ Differences[bb, n], {n, 1, Ceiling[max/2]}]; dd = Diagonal[diff]; bc[n_, n_] = 0; bc[n_, m_] /; m < n := bc[n, m] = bc[n-1, m+1] - bc[n-1, m]; bc[n_, m_] /; m == n+1 := bc[n, m] = -dd[[n+1]]; bc[n_, m_] /; m > n+1 := bc[n, m] = bc[n, m-1] + bc[n+1, m-1]; Table[bc[0, m], {m, 0, max}] // Numerator (* Jean-François Alcover, Aug 08 2012 *)

a(2*n+2)/a(2*n+1) = A000012(n)
BC(n, n) = 0, BC(n, n+1) = T(n, n+1) = T(n, n)/2 and BC(n, m) = BC(n-1, m+1) - BC(n-1, m); for the T(n, n+1) see A190339.
BC(0, m) = A191754(m)/A192366(m), i.e., the companion to the Bernoulli numbers.
Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A191754(k)/A192366(k). = (-1)^(n+1)*A191754(n)/ A192366(n).
Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A164555(k)/A027642(k). = (-1)^n*A164555(n)/A027642(n).
b(n) = A191754(n)/A192366(n) + A164555(n)/A027642(n) = [1, 1, 2/3, 1/3, 2/15, 1/15, 2/35, 1/35, -2/105, -1/105, ...] leads to b(2*n)/b(2*n+1) = 2 for n>1.

Edited by Johannes W. Meijer, Jul 02 2011

A309675 a(n) = 4^n^2 + n!.

Original entry on oeis.org

2, 5, 258, 262150, 4294967320, 1125899906842744, 4722366482869645214416, 316912650057057350374175806384, 340282366920938463463374607431768251776, 5846006549323611672814739330865132078623730534784, 1606938044258990275541962092341162602522202993782792838930176

Offset: 0

Andrew M. Kamal, Aug 11 2019

nonn

			a(1) = 5 since 1^1=1, (4^1) + 1! = 5;
a(2) = 4^2^2 = 4^4 = 256, 256 + 2! = 256 + 2*1 = 258.

Andrew M. Kamal, New Field

Cf. A309669, A192366.

cp's OEIS Frontend

A117575 Expansion of (1-x^3)/((1-x)(1+2x^2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A191754 Numerators of a companion to the Bernoulli numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A309675 a(n) = 4^n^2 + n!.

Views

Author

Keywords

Examples

Links

Crossrefs

cp's OEIS Frontend

A117575 Expansion of (1-x^3)/((1-x)*(1+2*x^2)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

SageMath

Formula

A191754 Numerators of a companion to the Bernoulli numbers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A309675 a(n) = 4^n^2 + n!.

Views

Author

Keywords

Examples

Links

Crossrefs

A117575 Expansion of (1-x^3)/((1-x)(1+2x^2)).