A045823 -id:A045823 - cp's OEIS frontend

A002408 Expansion of 8-dimensional cusp form.

0, 1, -8, 28, -64, 126, -224, 344, -512, 757, -1008, 1332, -1792, 2198, -2752, 3528, -4096, 4914, -6056, 6860, -8064, 9632, -10656, 12168, -14336, 15751, -17584, 20440, -22016, 24390, -28224, 29792, -32768, 37296, -39312, 43344, -48448, 50654, -54880, 61544, -64512, 68922

Offset: 0

N. J. A. Sloane and Mira Bernstein

Essentially the same as A007331.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
"For Gamma, it is known that any modular form is a weighted homogeneous polynomial in Theta_Z, which has weight 1/2, and the modular form delta_8(t) := e^(Pi i tau) Product_{m=1..oo} ((1 - e^(Pi i m tau)) (1 + e^(2 Pi i m tau)))^8 = e^(Pi i tau) - 8 e^(2 Pi i tau) + 28 e^(3 Pi i tau) - 64 e^(4 Pi i tau) + 126 e^(5 Pi i tau) ... of weight 4." [Elkies, p. 1242]

			G.f. = q - 8*q^2 + 28*q^3 - 64*q^4 + 126*q^5 - 224*q^6 + 344*q^7 ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 187.
Friedrich Hirzebruch, Thomas Berger, and Rainer Jung, Manifolds and Modular Forms, Vieweg 1994, p. 133.

T. D. Noe, Table of n, a(n) for n = 0..1000
Noam D. Elkies, Lattices, Linear Codes and Invariants, Part I, Notices of the Amer. Math. Soc., 47 (No. 10, Nov. 2000), 1238-1245, see p. 1242.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A007331, A010054, A045823, A121373, .

q*product((1-q^(2*k-1))^8*(1-q^(4*k))^8, k=1..75);

a[0] = 0; a[n_] := -(-1)^n*Sum[ Mod[n/d, 2]*d^3, {d, Divisors[n]}]; Table[a[n], {n, 0, 41}] (* Jean-François Alcover, Jan 27 2012, after Michael Somos *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^8, {q, 0, n}]; (* Michael Somos, May 25 2014 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (prod(k=1, n, (1 -( k%4==0) * x^k) * (1 - (k%2==1) * x^k), 1 + A))^8, n))}; /* Michael Somos, Jul 16 2004 */

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, (n/d%2) * d^3))}; /* Michael Somos, May 31 2005 */

from sympy import divisors
def a(n): return 0 if n == 0 else -(-1)**n * sum([((n//d)%2) * d**3 for d in divisors(n)])
print([a(n) for n in range(101)])  # Indranil Ghosh, Jun 24 2017

A = ModularForms( Gamma0(4), 4, prec=70) . basis(); A[1] - 8*A[2] # _Michael Somos, May 25 2014

Expansion of (eta(q)* eta(q^4) / eta(q^2))^8 in powers of q. - Michael Somos, Jul 16 2004
Euler transform of period 4 sequence [-8, 0, -8, -8, ...]. - Michael Somos, Jul 16 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = +u^4*w*v + 16*u^3*w*v^2 + 16*u^2*w^2*v^2 + 256*u^3*w^3 + 256*u^3*w^2*v + 4096*u^2*w^3*v + 4096*u*w^4*v + 4096*u*w^3*v^2 - u^2*v^4 - 16*u^2*w*v^3 - 256*u*w^2*v^3 - 256*w^2*v^4. - Michael Somos, May 31 2005
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^4*u6^4 + u1^3*u2*u3^3*u6 + 2*u1*u2^3*u3*u6^3 - u2^4*u3^4.
Expansion of q * psi(-q)^8 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Mar 20 2008
a(n) is multiplicative with a(2^e) = -8^e if e>0, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1). - Michael Somos, Mar 20 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 16 (t/i)^4 f(t) where q = exp(2 Pi i t).
G.f.: x * (Product_{k>0} (1 - x^(2*k-1)) * (1 - x^(4*k)))^8.
a(n) = -(-1)^n * A007331(n).
a(2*n) = -8 * A007331(n). a(2*n + 1) = A045823(n). - Michael Somos, May 25 2014
Dirichlet g.f.: zeta(s-3) * zeta(s) * (1 - 1/2^s) * (1 - 1/2^(s-4)). - Amiram Eldar, Nov 03 2023

A091986 a(0)=1, a(n) = sigma_3(2n).

Original entry on oeis.org

1, 9, 73, 252, 585, 1134, 2044, 3096, 4681, 6813, 9198, 11988, 16380, 19782, 25112, 31752, 37449, 44226, 55261, 61740, 73710, 86688, 97236, 109512, 131068, 141759, 160454, 183960, 201240, 219510, 257544, 268128, 299593, 335664, 358722, 390096, 442845

Offset: 0

N. J. A. Sloane, Mar 20 2004

Bisection of A001158.

Cf. A013662, A045823.

Join[{1},Table[DivisorSigma[3,n],{n,2,80,2}]]  (* Harvey P. Dale, Feb 26 2011 *)

a(n) = if(n < 1, 1, sigma(2*n, 3)); \\ Amiram Eldar, Dec 12 2023

Sum_{k=1..n} a(k) ~ (17*zeta(4)/8) * n^4. - Amiram Eldar, Dec 12 2023

G.f.: 1 + Sum_{k>=1} k^3*x^(lcm(k, 2)/2)/(1 - x^(lcm(k, 2)/2)). - Miles Wilson, Jul 10 2025

A092342 a(n) = sigma_3(3n+1).

Original entry on oeis.org

1, 73, 344, 1134, 2198, 4681, 6860, 11988, 15751, 25112, 29792, 44226, 50654, 73710, 79508, 109512, 117993, 160454, 167832, 219510, 226982, 299593, 300764, 390096, 389018, 500780, 493040, 620298, 619164, 779220, 756112, 934416, 912674, 1149823, 1092728

Offset: 0

N. J. A. Sloane, Mar 20 2004

			q + 73*q^4 + 344*q^7 + 1134*q^10 + 2198*q^13 + 4681*q^16 + ...

Trisection of A001158.

Cf. A000731, A013662, A033690, A045823, A091986, A092341, A092343.

DivisorSigma[3,3*Range[0,40]+1] (* Harvey P. Dale, Apr 22 2019 *)

{a(n) = if(n<0, 0, sigma(3*n+1, 3))} /* Michael Somos, Aug 22 2007 */

Expansion of q^(-1/3) * c(q) * (c(q)^3 + b(q)^3 / 3) in powers of q where b(), c() are cubic AGM functions. - Michael Somos, Aug 22 2007
If b(3*n) = 0, b(3*n+1) = a(n), b(3*n+2) = A092343(n), then b(n) is multiplicative with b(3^e) = 0^e, b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) otherwise. - Michael Somos, Aug 22 2007
a(n) = A000731(n) + 81*A033690(n-1). - Michael Somos, Aug 22 2007
Sum_{k=0..n} a(k) ~ (20*zeta(4)/3) * n^4. - Amiram Eldar, Dec 12 2023

A092341 a(0)=1, a(n) = sigma_3(3n).

Original entry on oeis.org

1, 28, 252, 757, 2044, 3528, 6813, 9632, 16380, 20440, 31752, 37296, 55261, 61544, 86688, 95382, 131068, 137592, 183960, 192080, 257544, 260408, 335664, 340704, 442845, 441028, 553896, 551881, 703136, 682920, 858438, 834176, 1048572, 1008324, 1238328, 1213632

Offset: 0

N. J. A. Sloane, Mar 20 2004

Trisection of A001158.

Cf. A013662, A045823, A091986, A092342, A092343.

Join[{1},DivisorSigma[3,3*Range[40]]] (* Harvey P. Dale, Feb 02 2012 *)

a(n) = if(n < 1, 1, sigma(3*n, 3)); \\ Amiram Eldar, Dec 12 2023

Sum_{k=1..n} a(k) ~ (83*zeta(4)/12) * n^4. - Amiram Eldar, Dec 12 2023

A092343 a(n) = sigma_3(3n+2).

Original entry on oeis.org

9, 126, 585, 1332, 3096, 4914, 9198, 12168, 19782, 24390, 37449, 43344, 61740, 68922, 97236, 103824, 141759, 148878, 201240, 205380, 268128, 276948, 358722, 357912, 455886, 458208, 589806, 571788, 715572, 704970, 888264, 864360, 1061937, 1030302, 1285830

Offset: 0

N. J. A. Sloane, Mar 20 2004

			G.f. = 9 + 126*x + 585*x^2 + 1332*x^3 + 3096*x^4 + 4914*x^5 + 9198*x^6 + 12168*x^7 + ...
G.f. = 9*q^2 + 126*q^5 + 585*q^8 + 1332*q^11 + 3096*q^14 + 4914*q^17 + 9198*q^20 + ...

Trisection of A001158.

Cf. A013662, A045823, A091986, A092341, A092342, A144614.

Table[DivisorSigma[3,3n+2],{n,0,40}] (* Harvey P. Dale, Jul 02 2011 *)

{a(n) = if( n<0, 0, sigma( 3*n + 2, 3))}; /* Michael Somos, May 30 2012 */

Expansion of q^(-2/3) * (a(q) * c(q))^2 in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, May 30 2012
Convolution square of A144614. - Michael Somos, May 30 2012
Sum_{k=0..n} a(k) ~ (20*zeta(4)/3) * n^4. - Amiram Eldar, Dec 12 2023

A135828 Expansion of psi(x^2)^8 * (psi(x)^8 + psi(-x)^8) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 36, 378, 2200, 8955, 28836, 78558, 188568, 410805, 828080, 1564686, 2804976, 4809370, 7927380, 12643560, 19594632, 29568204, 43626708, 63094550, 89501040, 124916931, 171803652, 232822908, 311683680, 412601490, 539849556, 699657642, 898801400, 1143680535

Offset: 0

Michael Somos, Nov 29 2007

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 36*x + 378*x^2 + 2200*x^3 + 8955*x^4 + 28836*x^6 + 78558*x^7 + ...
G.f. = q^3 + 36*q^5 + 378*q^7 + 2200*q^9 + 8955*q^11 + 28836*q^13 + 78558*q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A007331, A008774, A045823.

Basis( ModularForms( Gamma1(4), 8), 60)[4]; /* Michael Somos, Oct 15 2015 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x]^8 (EllipticTheta[ 2, 0, x^(1/2)]^8 + EllipticTheta[ 2, Pi/4, x^(1/2)]^8 16) / 131072, {x, 0, 2 n + 3}]; (* Michael Somos, Oct 15 2015 *)

{a(n) = my(A); if( n<0, 0, n *= 2; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^24 + eta(x + A)^16 * eta(x^4 + A)^8 ) / ( 2 * eta(x + A)^8 * eta(x^2 + A)^16 / eta(x^4 + A)^16 ), n))};

Expansion of q^(-3) * ( eta(q^2)^24 + eta(q)^16 * eta(q^4)^8 ) / ( 2 * eta(q)^8 * eta(q^2)^16 / eta(q^4)^16 ) in powers of q^2.
7680 * a(n) = A008774(2*n + 3).
Convolution of A007331 and A045823.

A045819 Theta series of E_8 lattice with respect to midpoint of edge.

Original entry on oeis.org

2, 56, 252, 688, 1514, 2664, 4396, 7056, 9828, 13720, 19264, 24336, 31502, 40880, 48780, 59584, 74592, 86688, 101308, 123088, 137844, 159016, 190764, 207648, 235986, 275184, 297756, 335664, 384160, 410760, 453964, 520816, 553896, 601528

Offset: 0

N. J. A. Sloane

			2*q^(1/2) + 56*q^(3/2) + 252*q^(5/2) + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 123.

Gabriele Nebe and N. J. A. Sloane, Home page for this lattice.

Cf. A001158, A008438, A013662, A045823.

a[n_] := 2 DivisorSigma[3, 2 n + 1]; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jul 06 2017, after Benoit Cloitre *)

G.f.: (1/2)*(theta_2^2*theta_3^6 + theta_2^6*theta_3^2).
a(n) = 2*sigma_3(2n+1). - Benoit Cloitre, Apr 12 2003
a(n) = 2 * A045823(n). - Alois P. Heinz, Mar 21 2021
Sum_{k=0..n} a(k) ~ (15*zeta(4)/4) * n^4. - Amiram Eldar, Dec 12 2023

More terms from Benoit Cloitre, Apr 12 2003

A081861 a(n) = (1/24)(sigma_3(2n-1) - sigma_1(2*n-1)).

Original entry on oeis.org

0, 1, 5, 14, 31, 55, 91, 146, 204, 285, 400, 506, 655, 850, 1015, 1240, 1552, 1804, 2109, 2562, 2870, 3311, 3971, 4324, 4914, 5730, 6201, 6990, 8000, 8555, 9455, 10846, 11536, 12529, 14192, 14910, 16206, 18371, 19088, 20540, 22990, 23821, 25794

Offset: 1

Benoit Cloitre, Apr 11 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A000203, A001158, A008438, A045823, A081863(1).

Table[(DivisorSigma[3,2n-1]-DivisorSigma[1,2n-1])/24,{n,50}] (* Harvey P. Dale, Dec 15 2012 *)

a(n) = my(f = factor(2*n-1)); (sigma(f, 3) - sigma(f)) / 24; \\ Amiram Eldar, Jan 04 2025

a(n) = (A045823(n-1) - A008438(n-1)) / 24. - Amiram Eldar, Jan 04 2025

A204386 Expansion of (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.

Original entry on oeis.org

1, 12, 28, 96, 126, 336, 344, 768, 757, 1512, 1332, 2688, 2198, 4128, 3528, 6144, 4914, 9084, 6860, 12096, 9632, 15984, 12168, 21504, 15751, 26376, 20440, 33024, 24390, 42336, 29792, 49152, 37296, 58968, 43344, 72672, 50654, 82320, 61544, 96768

Offset: 1

Michael Somos, Jan 15 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			x + 12*x^2 + 28*x^3 + 96*x^4 + 126*x^5 + 336*x^6 + 344*x^7 + 768*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004018, A007331, A045823, A050468.

Cf. A000122, A000700, A010054, A121373.

a[n_]:= SeriesCoefficient[(EllipticTheta[2, 0, q^(1/2)]^8 + 4*EllipticTheta[2, 0, q]^8)/256, {q, 0, n}];  Table[a[n], {n,1,50}] (* G. C. Greubel, Apr 13 2018 *)
CoefficientList[Series[(EllipticTheta[2,0,q^(1/2)]^8 +4*EllipticTheta[2, 0, q]^8)/ 256, {q, 0, 50}], q] (* Vaclav Kotesovec, Apr 13 2018 *)

{a(n) = if( n<1, 0, if( n%2, sigma( n, 3), 12 * sumdiv( n/2, d, (n/2/d%2) * d^3)))}

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / eta(x + A))^8 + 4 * x * (eta(x^4 + A)^2 / eta(x^2 + A))^8, n))}

Expansion of x * psi(x)^8 + 4 * x^2 * psi(x^2)^8 in powers of x where psi() is a Ramanujan theta function.
Expansion of (eta(q^2)^2 / eta(q))^8 + 4 * (eta(q^4)^2 / eta(q^2))^8 in powers of q.
a(n) is multiplicative with a(2^e) = 3/2 * 8^e if e>0, a(p^e) = ((p^3) ^ (e+1) - 1) / (p^3 - 1).
a(2*n + 1) = A045823(n). a(2*n) = 12 * A007331(n).
Convolution of this sequence with A004018 is A050468.
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 + 1/2^(s-2)) * (1 - 1/2^s) * zeta(s-3) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^4, where c = 5*Pi^4/1536 = 0.317086... . (End)

A291124 Expansion of phi(x)^6 * phi(-x)^2 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 8, 16, -32, -144, -16, 448, 192, -912, -88, 2016, -352, -4032, 176, 5504, 64, -7056, 400, 12112, 352, -18144, -768, 21312, -448, -25536, -968, 35168, 1216, -49536, 1584, 56448, -1280, -56208, 1408, 78624, -384, -109008, -1296, 109760, -704, -114912, -1584

Offset: 0

Michael Somos, Aug 17 2017

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700)

			G.f. = 1 + 8*x + 16*x^2 - 32*x^3 - 144*x^4 - 16*x^5 + 448*x^6 + 192*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A030211, A045820, A045823, A207541.

A := Basis( ModularForms( Gamma0(16), 4), 42); A[1] + 8*A[2] + 16*A[3] - 32*A[4] - 144*A[5] - 16*A[6] + 448*A[7] + 192*A[8] - 912*A[9];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6 EllipticTheta[ 4, 0, x]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[x^2]^7 / (QPochhammer[ x]^2 QPochhammer[ x^4]^3))^4, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3))^4, n))};

lista(nn) = {q='q+O('q^nn); Vec((eta(q^2)^7/(eta(q)^2*eta(q^4)^3))^4)} \\ Altug Alkan, Mar 21 2018

Expansion of (eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3))^4 in powers of q.
Euler transform of period 4 sequence [8, -20, 8, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 512 (t/i)^4 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A045820.
G.f.: Product_{k>0} (1 - x^(2*k))^28 / ((1 - x^k)^8 * (1 - x^(4*k))^12).
a(2*n + 1) = 8 * A030211(n). a(4*n + 2) = 16 * A045823(n).
a(2*n) = 16 * (-1)^n * (-sigma_3(n) + sigma_3(n/4)) where sigma_3(n) is the sum of the cubes of the divisors of n if n is an integer else 0.
Convolution square of A207541.

cp's OEIS Frontend

A002408 Expansion of 8-dimensional cusp form.

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A091986 a(0)=1, a(n) = sigma_3(2n).

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A092342 a(n) = sigma_3(3n+1).

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A092341 a(0)=1, a(n) = sigma_3(3n).

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A092343 a(n) = sigma_3(3n+2).

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A135828 Expansion of psi(x^2)^8 * (psi(x)^8 + psi(-x)^8) / 2 in powers of x^2 where psi() is a Ramanujan theta function.

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A045819 Theta series of E_8 lattice with respect to midpoint of edge.

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A081861 a(n) = (1/24)(sigma_3(2n-1) - sigma_1(2*n-1)).