A099587 -id:A099587 - cp's OEIS frontend

A099586 Constant term in (1+x)^n mod (1+x^4).

1, 1, 1, 0, -4, -14, -34, -68, -116, -164, -164, 0, 560, 1912, 4616, 9232, 15760, 22288, 22288, 0, -76096, -259808, -627232, -1254464, -2141504, -3028544, -3028544, 0, 10340096, 35303296, 85229696, 170459392, 290992384, 411525376, 411525376, 0, -1405035520

Offset: 1

Ralf Stephan, Oct 24 2004

Equals real part of term (1,1) in M^n, where M = a 2 X 2 matrix [1,1; i,1], where i = sqrt(-1). - Gary W. Adamson, Mar 25 2009
{A099586, A099587, A099588, A099589} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x)} of order 4.
For the definition, see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jul 03 2017
a(n) = 0 if and only if n == 4 (mod 8). - Robert Israel, Jul 04 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

Colin Barker, Table of n, a(n) for n = 1..1000
John B. Dobson, A matrix variation on Ramus's identity for lacunary sums of binomial coefficients, arXiv preprint arXiv:1610.09361 [math.NT], 2016.
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Vladimir Shevelev, Coefficient of x^k in ((x+1)^n modulo x^N+1), seqfan, Thu Jul 20 2017.
Gregory Tollisen and Tamás Lengyel, A congruential identity and the 2-adic order of lacunary sums of binomial coefficients, Integers 4 (2004), #A4.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-2).

Cf. A099587, A099588, A099589.

seq(eval(rem((1+x)^n, 1+x^4,x),x=0),n=1..40); # Robert Israel, Jul 03 2017

RecurrenceTable[{a[n] == 4 * a[n - 1] - 6 * a[n - 2] + 4 * a[n - 3] - 2 * a[n - 4], a[1] = 1, a[2] = 1, a[3] = 1, a[4] = 0}, a, {n, 50}] (* G. C. Greubel, Nov 10 2015 *)
a[n_] := HypergeometricPFQ[{(1-n)/4, (2-n)/4, (3-n)/4, -n/4}, {1/4, 1/2, 3/4}, -1]; Array[a, 40] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)

PARI
```
a(n) = polcoeff(((1+x)^n)%(x^4+1),0)
```

Vec(-x*(2*x-1)*(x^2-x+1)/(2*x^4-4*x^3+6*x^2-4*x+1) + O(x^100)) \\ Colin Barker, Nov 08 2015

a(n) = real(([1,1; I,1])^n)[1,1]; \\ Michel Marcus, Nov 08 2015

From Colin Barker, Nov 08 2015: (Start)
a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-2*a(n-4) for n>4.
G.f.: -x*(2*x-1)*(x^2-x+1) / (2*x^4-4*x^3+6*x^2-4*x+1).
(End)
a(n) = (1/2)*((2+sqrt(2))^(n/2)*cos(n*Pi/8) + (2-sqrt(2))^(n/2)*cos(3*n*Pi/8)). - G. C. Greubel, Nov 10 2015
From Vladimir Shevelev, Jun 29 2017: (Start)
a(n) = Sum_{k >= 0}(-1)^k*binomial(n,4*k).
a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n)/8)/2), where round(x) is the integer nearest to x.
a(n+m) = a(n)*a(m) - K_4(n)*K_2(m) - K_3(n)*K_3(m) - K_2(n)*K_4(m), where K_2 is A099587, K_3 is A099588 and K_4 is A099589.
(End)

A099589 Expansion of x^3 / (1 - 4x + 6x^2 - 4x^3 + 2x^4).

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 20, 34, 48, 48, 0, -164, -560, -1352, -2704, -4616, -6528, -6528, 0, 22288, 76096, 183712, 367424, 627232, 887040, 887040, 0, -3028544, -10340096, -24963200, -49926400, -85229696, -120532992, -120532992, 0, 411525376, 1405035520, 3392055808

Offset: 0

Ralf Stephan, Oct 24 2004

{A099586, A099587, A099588, A099589} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x)} of order 4. For the definition, see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jul 04 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Vladimir Shevelev, Coefficient of x^k in ((x+1)^n modulo x^N+1), seqfan, Thu Jul 20 2017.
G. Tollisen and T. Lengyel, A congruential identity and the 2-adic order of lacunary sums of binomial coefficients, Integers 4 (2004), #A4.
Maran van Heesch, The multiplicative complexity of symmetric functions over a field with characteristic p, Thesis, 2014.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-2).

Cf. A099586, A099587, A099588.

Round@Table[(1/(2*Sqrt[2]))*((2-Sqrt[2])^(n/2)*(Cos[3*Pi*n/8] + Sin[3*Pi*n/8]) + (2+Sqrt[2])^(n/2)*(Sin[Pi*n/8] - Cos[Pi*n/8])), {n, 0, 40}] (* G. C. Greubel, Nov 07 2015 *)
RecurrenceTable[{a[n] == 4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4], a[0]==0, a[1]==0, a[2]==0, a[3]==1}, a, {n, 0, 40}] (* G. C. Greubel, Nov 10 2015 *)
Table[Sum[(-1)^k*Binomial[n, 4 k + 3], {k, 0, n}], {n, 0, 37}] (* Michael De Vlieger, Jun 30 2017 *)
a[n_] := n*(n-1)*(n-2)/6 HypergeometricPFQ[{(3-n)/4, (4-n)/4, (5-n)/4, (6-n)/4}, {5/4, 3/2, 7/4}, -1]; Array[a, 40, 0] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)

PARI
```
a(n) = polcoeff(((1+x)^n)%(x^4+1),3)
```

concat([0, 0], Vec(x^3/((1-x)^4+x^4) + O(x^50))) \\ Altug Alkan, Nov 08 2015

a(n) = sum(t=0, (n-3)\4, (-1)^t*binomial(n,4*t+3)); \\ Michel Marcus, Jun 30 2017

G.f.: x^3/((1-x)^4 + x^4), the binomial transform of x^3/(1+x^4). - Paul Barry, Apr 01 2005
Coefficient of x^3 in (1+x)^n mod (1 + x^4).
a(n) = (1/(2*sqrt(2)))*((2-sqrt(2))^(n/2)*(cos(3*Pi*n/8) + sin(3*Pi*n/8)) + (2+sqrt(2))^(n/2)*(sin(Pi*n/8) - cos(Pi*n/8))). - Paul Barry, Apr 01 2005
From Colin Barker, Nov 08 2015: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-4) for n > 4.
G.f.: x^3 / (2*x^4 - 4*x^3 + 6*x^2 - 4*x + 1). (End)
From Vladimir Shevelev, Jul 04 2017: (Start)
a(n) = Sum_{t >= 0} (-1)^t*binomial(n,4*t+3).
a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-6)/8)/2), where round(x) is the integer nearest to x.
a(n+m) = a(n)*K_1(m) + K_3(n)*K_2(m) + K_2(n)*K_3(m) + K_1(n)*a(m), where
K_1 is A099586, K_2 is A099587, K_3 is A099588. (End)

a(0)=0 added by N. J. A. Sloane, Jul 04 2017

A099588 Coefficient of x^2 in (1+x)^n mod 1+x^4.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 14, 14, 0, -48, -164, -396, -792, -1352, -1912, -1912, 0, 6528, 22288, 53808, 107616, 183712, 259808, 259808, 0, -887040, -3028544, -7311552, -14623104, -24963200, -35303296, -35303296, 0, 120532992, 411525376, 993510144, 1987020288, 3392055808, 4797091328, 4797091328

Offset: 0

Ralf Stephan, Oct 24 2004

{A099586, A099587, A099588, A099589} is the difference analog of the trigonometric functions {k_1(x), k_2(x), k_3(x), k_4(x)} of order 4. For the definition, see [Erdelyi] and the Shevelev link. - Vladimir Shevelev, Jul 04 2017

A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, 1955, Vol. 3, Chapter XVIII.

G. C. Greubel, Table of n, a(n) for n = 0..1000
Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Vladimir Shevelev, Coefficient of x^k in ((x+1)^n modulo x^N+1), seqfan, Thu Jul 20 2017.
G. Tollisen and T. Lengyel, A congruential identity and the 2-adic order of lacunary sums of binomial coefficients, Integers 4 (2004), #A4.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-2).

Cf. A099586, A099587, A099589.

f:= rectoproc({rec, a(0)=0,a(1)=0,a(2)=1,a(3)=3},a(n),remember):
map(f, [$0..100]); # Robert Israel, Jun 30 2017

RecurrenceTable[{a[n]==4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4], a[1]=0, a[2]=1, a[3]=3, a[4]=6}, a, {n, 1, 200}] (* G. C. Greubel, Nov 10 2015 *)
Table[Sum[(-1)^k*Binomial[n, 4 k + 2], {k, 0, n}], {n, 0, 36}] (* Michael De Vlieger, Jun 30 2017 *)
a[n_] := n*(n-1)/2 HypergeometricPFQ[{(2-n)/4, (3-n)/4, (4-n)/4, (5-n)/4}, {3/4, 5/4, 3/2}, -1]; Array[a, 40, 0] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)

x='x+O('x^55); concat([0, 0], Vec(-x^2*(x-1)/(2*x^4-4*x^3+6*x^2-4*x+1))) \\ Altug Alkan, Nov 11 2015

a(n) = sum(t=0, (n-2)\4, (-1)^t*binomial(n,4*t+2)); \\ Michel Marcus, Jun 30 2017

a(n)=polcoeff(lift(Mod(1+x,1+x^4)^n),2); \\ Joerg Arndt, Feb 22 2018

G.f.: -x^2*(x-1) / (2*x^4-4*x^3+6*x^2-4*x+1). - Colin Barker, Jul 15 2013
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - 2*a(n-4). - G. C. Greubel, Nov 10 2015
a(n) = (1/2)*((2+sqrt(2))^(n/2) * sin(n*Pi/8) - (2-sqrt(2))^(n/2)*sin(3*n*Pi/8)). - G. C. Greubel, Nov 10 2015
From Vladimir Shevelev, Jul 04 2017 (Start)
a(n) = Sum_{k>=0}(-1)^k*binomial(n,4*k+2);
a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-4)/8)/2), where round(x) is the integer nearest to x;
a(n+m) = a(n)*K_1(m) + K_2(n)*K_2(m) + K_1(n)*a(m) - K_4(n)*K_4(m), where K_1 is A099586, K_2 is A099587, K_4 is A099589. (End)

a(0)=0 added by N. J. A. Sloane, Jul 04 2017

a(673) in b-file corrected by Andrew Howroyd, Feb 21 2018

A307079 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 0, 1, 1, 2, 3, 3, -4, 1, 1, 2, 3, 4, 0, -8, 1, 1, 2, 3, 4, 4, -9, -8, 1, 1, 2, 3, 4, 5, 0, -27, 0, 1, 1, 2, 3, 4, 5, 5, -14, -54, 16, 1, 1, 2, 3, 4, 5, 6, 0, -48, -81, 32, 1, 1, 2, 3, 4, 5, 6, 6, -20, -116, -81, 32, 1

Offset: 0

Seiichi Manyama, Mar 22 2019

			Square array begins:
   1,  1,   1,    1,   1,   1, 1, 1, 1, ...
   1,  2,   2,    2,   2,   2, 2, 2, 2, ...
   1,  2,   3,    3,   3,   3, 3, 3, 3, ...
   1,  0,   3,    4,   4,   4, 4, 4, 4, ...
   1, -4,   0,    4,   5,   5, 5, 5, 5, ...
   1, -8,  -9,    0,   5,   6, 6, 6, 6, ...
   1, -8, -27,  -14,   0,   6, 7, 7, 7, ...
   1,  0, -54,  -48, -20,   0, 7, 8, 8, ...
   1, 16, -81, -116, -75, -27, 0, 8, 9, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns 1-6 give A000012, A099087, A057682(n+1), A099587(n+1), A289321(n+1), A307089.

Cf. A306914, A307039, A307078.

T[n_, k_] := Sum[(-1)^j * Binomial[n+1, k*j+1], {j, 0, Floor[n/k]}]; Table[T[n-k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 2021 *)

A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+1,k*j+1).

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i,k*j) * binomial(n-i,k*j).

A290286 Determinant of circulant matrix of order 4 with entries in the first row (-1)^jSum_{k>=0}(-1)^kbinomial(n, 4*k+j), j=0,1,2,3.

Original entry on oeis.org

1, 0, 0, 0, -1008, -37120, -473600, 0, 63996160, 702013440, 2893578240, 0, -393379835904, -12971004067840, -160377313820672, 0, 21792325059543040, 239501351489372160, 987061897553510400, 0, -134124249770961666048, -4422152303189489090560

Offset: 0

Vladimir Shevelev and Peter J. C. Moses, Jul 26 2017

sign

In the Shevelev link the author proved that, for odd N>=3 and every n>=1, the determinant of circulant matrix of order N with entries in the first row (-1)^j*Sum{k>=0}(-1)^k*binomial(n, N*k+j), j=0..N-1, is 0.

This sequence shows what happens for the first even N>3.

Vladimir Shevelev, Combinatorial identities generated by difference analogs of hyperbolic and trigonometric functions of order n, arXiv:1706.01454 [math.CO], 2017.
Wikipedia, Circulant matrix

Cf. A099586 (prefixed by a(0)=1), A099587, A099588, A099589, A290285.

seq(LinearAlgebra:-Determinant(Matrix(4,shape=Circulant[seq((-1)^j*
add((-1)^k*binomial(n, 4*k+j),k=0..n/4),j=0..3)])),n=0..50); # Robert Israel, Jul 26 2017

ro[n_] := Table[Sum[(-1)^(j+k) Binomial[n, 4k+j], {k, 0, n/4}], {j, 0, 3}];
M[n_] := Table[RotateRight[ro[n], m], {m, 0, 3}];
a[n_] := Det[M[n]];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 09 2018 *)

from sympy.matrices import Matrix
from sympy import binomial
def mj(j, n): return (-1)**j*sum((-1)**k*binomial(n, 4*k + j) for k in range(n//4 + 1))
def a(n):
    m=Matrix(4, 4, lambda i,j: mj((i-j)%4,n))
    return m.det()
print([a(n) for n in range(22)]) # Indranil Ghosh, Jul 31 2017

a(n) = 0 for n == 3 (mod 4).

G.f. (empirical): (1/8)*(68*x^2+1)/(16*x^4+136*x^2+1)+(1/4)*(68*x^2-8*x+1)/(16*x^4+64*x^3+128*x^2-16*x+1)+(1/2)*(12*x^2+1)/(16*x^4+24*x^2+1)+3/(8*(4*x^2+1))-(1/4)*(12*x^2-4*x+1)/(16*x^4-32*x^3+32*x^2-8*x+1)-(1/4)*(4*x^2+1)/(16*x^4+1)+(1/4)*(12*x^2+4*x+1)/(16*x^4+32*x^3+32*x^2+8*x+1). - Robert Israel, Jul 26 2017

A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2j) binomial(n-k,2*j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -2, -2, 1, 1, 1, 1, -5, -8, -5, 1, 1, 1, 1, -9, -17, -17, -9, 1, 1, 1, 1, -14, -29, -34, -29, -14, 1, 1, 1, 1, -20, -44, -54, -54, -44, -20, 1, 1, 1, 1, -27, -62, -74, -74, -74, -62, -27, 1, 1, 1, 1, -35, -83, -90, -74, -74, -90, -83, -35, 1, 1

Offset: 0

Seiichi Manyama, Mar 24 2019

			Triangle begins:
n\k | 0  1    2    3    4    5    6  7  8
----+-------------------------------------
0   | 1;
1   | 1, 1;
2   | 1, 1,   1;
3   | 1, 1,   1,   1;
4   | 1, 1,   0,   1,   1;
5   | 1, 1,  -2,  -2,   1,   1;
6   | 1, 1,  -5,  -8,  -5,   1,   1;
7   | 1, 1,  -9, -17, -17,  -9,   1, 1;
8   | 1, 1, -14, -29, -34, -29, -14, 1, 1;

Seiichi Manyama, Rows n = 0..139, flattened

Row sums give A099587(n+1).
T(2*n,n) gives A307091.
Cf. A007318, A098593, A119326, A119335.

T[n_, k_] := Sum[(-1)^j * Binomial[k, 2*j] * Binomial[n - k, 2*j], {j, 0, n - k}]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, May 20 2021 *)

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A099586 Constant term in (1+x)^n mod (1+x^4).

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A099589 Expansion of x^3 / (1 - 4*x + 6*x^2 - 4*x^3 + 2*x^4).

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A099588 Coefficient of x^2 in (1+x)^n mod 1+x^4.

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A307079 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-2))/((1-x)^k+x^k).

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A290286 Determinant of circulant matrix of order 4 with entries in the first row (-1)^j*Sum_{k>=0}(-1)^k*binomial(n, 4*k+j), j=0,1,2,3.

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A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2*j) * binomial(n-k,2*j).

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A099589 Expansion of x^3 / (1 - 4x + 6x^2 - 4x^3 + 2x^4).

A290286 Determinant of circulant matrix of order 4 with entries in the first row (-1)^jSum_{k>=0}(-1)^kbinomial(n, 4*k+j), j=0,1,2,3.

A307090 Number triangle T(n,k) = Sum_{j=0..n-k} (-1)^j * binomial(k,2j) binomial(n-k,2*j).