A099589 -id:A099589 - cp's OEIS frontend

A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k).

1, 1, 0, 1, 2, 0, 1, 3, 2, 0, 1, 4, 6, 0, 0, 1, 5, 10, 9, -4, 0, 1, 6, 15, 20, 9, -8, 0, 1, 7, 21, 35, 34, 0, -8, 0, 1, 8, 28, 56, 70, 48, -27, 0, 0, 1, 9, 36, 84, 126, 125, 48, -81, 16, 0, 1, 10, 45, 120, 210, 252, 200, 0, -162, 32, 0

Offset: 0

Seiichi Manyama, Mar 16 2019

			Square array begins:
   1,  1,    1,    1,   1,    1,    1,    1, ...
   0,  2,    3,    4,   5,    6,    7,    8, ...
   0,  2,    6,   10,  15,   21,   28,   36, ...
   0,  0,    9,   20,  35,   56,   84,  120, ...
   0, -4,    9,   34,  70,  126,  210,  330, ...
   0, -8,    0,   48, 125,  252,  462,  792, ...
   0, -8,  -27,   48, 200,  461,  924, 1716, ...
   0,  0,  -81,    0, 275,  780, 1715, 3432, ...
   0, 16, -162, -164, 275, 1209, 2989, 6434, ...

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

Columns 1-9 give A000007, A099087, A057083, A099589(n+3), A289389(n+4), A306940, (-1)^n * A049018(n), A306941, A306942.

Cf. A039912, A306913, A306915, A307039, A307079, A307394.

A[n_, k_] := SeriesCoefficient[1/((1-x)^k + x^k), {x, 0, n}];
Table[A[n-k+1, k], {n, 0, 11}, {k, n+1, 1, -1}] // Flatten (* Jean-François Alcover, Mar 20 2019 *)

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

A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+k-1,k*j+k-1) * binomial(n-i+k-1,k*j+k-1). - Seiichi Manyama, Apr 07 2019

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

Original entry on oeis.org

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)

A099587 a(n) = coefficient of x in (1+x)^n mod (1+x^4).

Original entry on oeis.org

0, 1, 2, 3, 4, 4, 0, -14, -48, -116, -232, -396, -560, -560, 0, 1912, 6528, 15760, 31520, 53808, 76096, 76096, 0, -259808, -887040, -2141504, -4283008, -7311552, -10340096, -10340096, 0, 35303296, 120532992, 290992384, 581984768

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 03 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, A099588, A099589.

Cf. A057681, A099586, A139398, A133476, A139714, A139748, A139761.

RecurrenceTable[{a[1]=1, a[2]=2, a[3]=3, a[4]=4, a[n] = 4*a[n-1] - 6*a[n-2] + 4*a[n-3] - 2*a[n-4]}, a, {n, 1, 100}] (* G. C. Greubel, Nov 09 2015 *)
a[n_] := n*HypergeometricPFQ[{(1-n)/4, (2-n)/4, (3-n)/4, (4-n)/4}, {1/2, 3/4, 5/4}, -1]; Array[a, 40, 0] (* Jean-François Alcover, Jul 20 2017, from Vladimir Shevelev's first formula *)
LinearRecurrence[{4,-6,4,-2},{0,1,2,3},50] (* Harvey P. Dale, Mar 27 2022 *)

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

G.f.: x*(x-1)^2 / (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 09 2015
From Vladimir Shevelev, Jun 29 2017: (Start)
a(n) = Sum_{k >= 0}(-1)^k*binomial(n,4*k+1).
a(n) = round((2+sqrt(2))^(n/2)*cos(Pi*(n-2)/8)/2), where round(x) is the integer nearest to x.
a(n+m) = a(n)*K_1(m) + K_1(n)*a(m) - K_4(n)*K_3(m) - K_3(n)*K_4(m), where K_1 is A099586, K_3=A099588, and K_4=A099589.
(End)
a(n) = A099589(n+2)-2*A099589(n+1)+A099589(n). - R. J. Mathar, Jun 28 2025

a(0)=0 added by N. J. A. Sloane, Jun 30 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

A307394 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((1-x)^k+x^k).

Original entry on oeis.org

1, 1, 3, 1, 4, 6, 1, 4, 9, 10, 1, 4, 10, 14, 15, 1, 4, 10, 19, 15, 21, 1, 4, 10, 20, 28, 8, 28, 1, 4, 10, 20, 34, 28, -7, 36, 1, 4, 10, 20, 35, 48, 1, -22, 45, 1, 4, 10, 20, 35, 55, 48, -80, -21, 55, 1, 4, 10, 20, 35, 56, 75, 0, -242, 12, 66, 1, 4, 10, 20, 35, 56, 83, 75, -164, -485, 77, 78

Offset: 0

Seiichi Manyama, Apr 07 2019

			Square array begins:
    1,   1,    1,    1,  1,   1,   1,   1,   1, ...
    3,   4,    4,    4,  4,   4,   4,   4,   4, ...
    6,   9,   10,   10, 10,  10,  10,  10,  10, ...
   10,  14,   19,   20, 20,  20,  20,  20,  20, ...
   15,  15,   28,   34, 35,  35,  35,  35,  35, ...
   21,   8,   28,   48, 55,  56,  56,  56,  56, ...
   28,  -7,    1,   48, 75,  83,  84,  84,  84, ...
   36, -22,  -80,    0, 75, 110, 119, 120, 120, ...
   45, -21, -242, -164,  0, 110, 154, 164, 165, ...

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

Columns 1-5 give A000217(n+1), A279230, A307395, A099589(n+3), A289388(n+3).

Cf. A306914, A307039, A307079, A307393.

T[n_, k_] := Sum[(-1)^j * Binomial[n+3, k*j + 3], {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+3,k*j+3).

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

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

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A306914 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/((1-x)^k+x^k).

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

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A099587 a(n) = coefficient of x in (1+x)^n mod (1+x^4).

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

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A307394 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. ((1-x)^(k-4))/((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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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.