A115979 -id:A115979 - cp's OEIS frontend

A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

1, 1, 0, 2, 1, 0, 2, 0, 0, 2, 2, 0, 1, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 2, 0

Offset: 0

N. J. A. Sloane

nonn

Number of solutions of 8*n + 4 = x^2 + 3*y^2 in positive odd integers. - Michael Somos, Sep 18 2004
Half the number of integer solutions of 4*n + 2 = x^2 + y^2 + z^2 where 0 = x + y + z and x and y are odd. - Michael Somos, Jul 03 2011
Given g.f. A(x), then q^(1/2) * 2 * A(q) is denoted phi_1(z) where q = exp(Pi i z) in Conway and Sloane.
Half of theta series of planar hexagonal lattice (A2) with respect to an edge.
Bisection of A002324. Number of ways of writing n as a sum of a triangular plus three times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 + x + 2*x^3 + x^4 + 2*x^6 + 2*x^9 + 2*x^10 + x^12 + x^13 + 2*x^15 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 2*q^13 + 2*q^19 + 2*q^21 + q^25 + q^27 + 2*q^31 + ...
a(6) = 2 since 8*6 + 4 = 52 = 5^2 + 3*3^2 = 7^2 + 3*1^2.

Burce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 223 Entry 3(i).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 103. See Eq. (13).
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Michael D. Hirschhorn, Three classical results on representations of a number, Sem. Lotharingien de Combinat. S42 (1999), B42f.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions, 2010.
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002324, A004016, A005881, A035178, A091393, A093766, A093829, A096936, A112298, A113447, A113661, A113974, A115979, A122860, A123331, A123484, A136748, A137608.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 202); A[2] + A[4]; /* Michael Somos, Jul 25 2014 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Jul 03 2011 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^2/(QP[q]*QP[q^3]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(2 n + 1))]; (* Michael Somos, Mar 06 2016 *)
%t A033762 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}]; (* Michael Somos, Mar 06 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 18 2004 */

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -12, d) * (n / d % 2)))}; /* Michael Somos, Nov 04 2005 */

{a(n) = if( n<0, 0, n = 8*n + 4; sum( j=1, sqrtint( n\3), (j%2) * issquare(n - 3*j^2)))} /* Michael Somos, Nov 04 2005 */

{a(n) = if( n<0, 0, sumdiv(2*n + 1, d, kronecker(-3, d)))}; /* Michael Somos, Mar 06 2016 */

Expansion of q^(-1/2) * (eta(q^2) * eta(q^6))^2 / (eta(q) * eta(q^3)) in powers of q. - Michael Somos, Apr 18 2004
Expansion of q^(-1) * (a(q) - a(q^4)) / 6 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Oct 24 2006
Expansion of psi(x) * psi(x^3) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 03 2011
Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, -2, ...]. - Michael Somos, Apr 18 2004
From Michael Somos, Sep 18 2004: (Start)
Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 4*u*v*w + 16*v*w^2 - 8*w*v^2 - w*u^2.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p==5 (mod 6) otherwise b(p^e) = e+1. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x * A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)
G.f.: (Sum_{j>0} x^((j^2 - j) / 2)) * (Sum_{k>0} x^(3(k^2 - k) / 2)) = Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k * (1 - x^k) * (1 - x^(4*k)) * (1 - x^(5*k)) / (1 - x^(12*k)). (End)
G.f.: s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - x^(4*k) / (1 + x^(4*k) + x^(8*k)). - Michael Somos, Nov 04 2005
a(n) = A002324(2*n + 1) = A035178(2*n + 1) = A091393(2*n + 1) = A093829(2*n + 1) = A096936(2*n + 1) = A112298(2*n + 1) = A113447(2*n + 1) = A113661(2*n + 1) = A113974(2*n + 1) = A115979(2*n + 1) = A122860(2*n + 1) = A123331(2*n + 1) = A123484(2*n + 1) = A136748(2*n + 1) = A137608(2*n + 1). A005881(n) = 2*a(n).
6 * a(n) = A004016(6*n + 3). - Michael Somos, Mar 06 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 23 2023

Corrected by Charles R Greathouse IV, Sep 02 2009

A122861 Expansion of phi(-q)chi(-q)psi(q^3) in powers of q where phi(),chi(),psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 2, 0, 2, -3, 2, 0, 1, -6, 2, 0, 2, 0, 2, 0, 3, -6, 0, 0, 2, -3, 2, 0, 2, -6, 2, 0, 0, 0, 4, 0, 2, -3, 2, 0, 2, -6, 0, 0, 1, -6, 2, 0, 4, 0, 2, 0, 0, -6, 2, 0, 2, 0, 2, 0, 3, -6, 2, 0, 2, 0, 0, 0, 2, -9, 2, 0, 0, -6, 2, 0, 4, 0, 2, 0, 2, 0, 0, 0, 2, -6, 4, 0, 0, -3, 4, 0, 0, -6, 2, 0, 2, 0, 2, 0, 1, -6, 0, 0, 4, -6, 2, 0, 2

Offset: 0

Michael Somos, Sep 15 2006

sign

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A097109, A097195, A115979, A129576.

Cf. A000122, A000700, A010054, A121373.

A122861[n_] := SeriesCoefficient[(QPochhammer[q]^3*QPochhammer[q^6]^2)/(QPochhammer[q^2]^2 *QPochhammer[q^3]), {q, 0, n}]; Table[A122861[n], {n, 0, 50}] (* G. C. Greubel, Oct 05 2017 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^6+A)^2/eta(x^2+A)^2/eta(x^3+A), n))}

{a(n)=local(A, p, e); if(n<0, 0, n=3*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 3*(e%2-1), if(p==3, 0, if(p%6==1, e+1, !(e%2)))))))}

Expansion of q^(-1/3)*eta(q)^3*eta(q^6)^2/(eta(q^2)^2*eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, -1, -2, -1, -3, -2, ...].
a(n) = b(3n+1) where b(n) is multiplicative and b(2^e) = -3(1+(-1)^e)/2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(4n+3) = 0.
a(n) = A115979(3n+1) = A097109(3n+1).
a(2n) = A097195(n) = A033687(2n); a(2n+1) = -3*A033687(2n+1).
a(n) = (-1)^n * A129576(n). - Amiram Eldar, Jan 28 2024

A096936 Half of number of integer solutions to the equation x^2 + 3y^2 = n.

Original entry on oeis.org

1, 0, 1, 3, 0, 0, 2, 0, 1, 0, 0, 3, 2, 0, 0, 3, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 6, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 6, 0, 0, 2, 0, 1, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0

Offset: 1

Michael Somos, Jul 15 2004

			G.f. = x + x^3 + 3*x^4 + 2*x^7 + x^9 + 3*x^12 + 2*x^13 + 3*x^16 + 2*x^19 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).

Antti Karttunen, Table of n, a(n) for n = 1..65537
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.

Cf. A033716, A093766, A115979.

sigmamr := proc(n,m,r) local a,d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d,m) = r then  a := a+1 ; end if; end do: a; end proc:
A002324 := proc(n) sigmamr(n,3,1)-sigmamr(n,3,2) ; end proc:
A096936 := proc(n) A002324(n) +2*(sigmamr(n,12,4)-sigmamr(n,12,8) ); end proc:
seq(A096936(n),n=1..90) ; # R. J. Mathar, Mar 23 2011

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1 || # == 3, 1, # == 2, 3 (1 + (-1)^#2)/2, Mod[#, 3] == 1, #2 + 1, True, (1 + (-1)^#2)/2] & @@@ FactorInteger[n])]; (* Michael Somos, Nov 20 2017 *)

{a(n) = if( n<1, 0, 1/2 * polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x*O(x^n)) * sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1 + x*O(x^n)), n))};

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 3], n)[n])}; /* Michael Somos, Jun 05 2005 */

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, p==2, 3 * (1 + (-1)^e) / 2, p%3==2, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Nov 20 2017 */

(definec (A096936 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n)) (rest (A096936 (A028234 n)))) (cond ((= 2 p) (* (if (odd? e) 0 3) rest)) ((= 3 p) rest) ((= 1 (modulo p 3)) (* (+ 1 e) rest)) (else (* (if (odd? e) 0 1) rest)))))) ;; With the memoization-macro definec, after the given multiplicative formula. - Antti Karttunen, Nov 20 2017

a(n) = A033716(n) / 2.
Multiplicative with a(2^e) = 3*(1+(-1)^e)/2, a(3^e) = 1, a(p^e) = (1+(-1)^e)/2 if p==2 (mod 3) and a(p^e) = 1+e if p==1 (mod 3). - Corrected by Antti Karttunen, Nov 20 2017
G.f.: ((Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(3*k^2)) - 1)/2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Oct 15 2022

A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, -2, 6, 0, 0, -4, 0, -2, 0, 0, 6, -4, 0, 0, 6, 0, 0, -4, 0, -4, 0, 0, 0, -2, 0, -2, 12, 0, 0, -4, 0, 0, 0, 0, 6, -4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, -6, 0, 0, 12, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0, -2, 12, 0, 0, -4, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, -8, 0, -4, 0, 0, 0, -4, 0, 0, 6, 0

Offset: 0

Michael Somos, Feb 09 2006

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 2*q - 2*q^3 + 6*q^4 - 4*q^7 - 2*q^9 + 6*q^12 - 4*q^13 + ...

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. A004016, A033716, A033762, A097195, A112604, A122861.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Nov 09 2013 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^2 / (eta(x^2 + A) * eta(x^6 + A)), n))}

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==2, -3 * ((e+1)%2), if( p==3, 1, if( p%6==1, e+1, (e+1)%2))))))} /* Michael Somos, Nov 09 2013 */

Expansion of theta_4(q) * theta_4(q^3) in powers of q.
Expansion of (4 * a(q^4) - a(q)) / 3 = (4 * b(q^4) - b(q)) * b(q) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM theta functions. - Michael Somos, Nov 09 2013
Expansion of (eta(q) * eta(q^3))^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -2, -1, -4, -1, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 2, 0, 6, 2, 0, -2, -6, 0, -2, 2, 0, ...]. - Michael Somos, Nov 09 2013
a(n) = -2*b(n) where b(n) is multiplicative and b(2^e) = -3 * (1 + (-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = 1+e if p == 1 (mod 6), b(p^e) = (1 +(-1)^e) / 2 if p == 5 (mod 6).
Given g.f. A(x), then B(x) = A(x)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v*(u + v)^2 - 4*u * (w^2 - v*w + v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033762. - Michael Somos, Nov 09 2013
G.f.: 1 - 2*(Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - 4 * x^(4*k) / (1 + x^(4*k) + x^(8*k))).
G.f.: (Sum_{k in Z} (-x)^(k^2)) * (Sum_{k in Z} (-x)^(3*k^2)).
a(n) = -2 * A115979(n) unless n=0. a(n) = (-1)^n * A033716(n).
a(3*n + 2) = a(4*n + 2) = 0. a(3*n) = a(n). a(2*n + 1) = -2 * A033762(n). a(3*n + 1) = -2 * A122861(n). a(4*n) = A004016(n). a(4*n + 1) = -2 * A112604(n). a(6*n + 1) = -2 * A097195(n). - Michael Somos, Nov 09 2013

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A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

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A122861 Expansion of phi(-q)chi(-q)psi(q^3) in powers of q where phi(),chi(),psi() are Ramanujan theta functions.

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A096936 Half of number of integer solutions to the equation x^2 + 3y^2 = n.

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A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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