A035154 -id:A035154 - cp's OEIS frontend

A008441 Number of ways of writing n as the sum of 2 triangular numbers.

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

Offset: 0

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). The present sequence gives the expansion coefficients of psi(q)^2.

Also the number of positive odd solutions to equation x^2 + y^2 = 8*n + 2. - Seiichi Manyama, May 28 2017

			G.f. = 1 + 2*x + x^2 + 2*x^3 + 2*x^4 + 3*x^6 + 2*x^7 + 2*x^9 + 2*x^10 + 2*x^11 + ...
G.f. for B(q) = q * A(q^4) = q + 2*q^5 + q^9 + 2*q^13 + 2*q^17 + 3*q^25 + 2*q^29 + 2*q^37 + 2*q^41 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag. See p. 139 Example (iv).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 102.
R. W. Gosper, Strip Mining in the Abandoned Orefields of Nineteenth Century Mathematics, in Computers in Mathematics (Ed. D. V. Chudnovsky and R. D. Jenks). New York: Dekker, 1990. See p. 279.
R. W. Gosper, Experiments and discoveries in q-trigonometry, in Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Editors: F. G. Garvan and M. E. H. Ismail. Kluwer, Dordrecht, Netherlands, 2001, pp. 79-105. [See Pi_q.]
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916. See vol. 2, p 31, Article 272.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991, p. 165.

T. D. Noe, Table of n, a(n) for n = 0..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 575, 16.23.1 and 16.23.2.
R. P. Agarwal, Lambert series and Ramanujan, Prod. Indian Acad. Sci. (Math. Sci.), v. 103, n. 3, 1993, pp. 269-293 (see p. 285).
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 72, Eq (31.2); p. 78, Eq. following (32.25).
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, 20 (1884), 97-167. See p. 108.
R. W. Gosper, Experiments and discoveries in q-trigonometry, Preprint.
R. W. Gosper, q-Trigonometry: Some Prefatory Afterthoughts.
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
Nicco, Conjectured identity of the product of two theta functions, Math Stackexchange, Sep 09 2015.
K. Ono, S. Robins and P. T. Wahl, On the representation of integers as sums of triangular numbers, Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp. 73-94.
Hjalmar Rosengren, Sums of triangular numbers from the Frobenius determinant, arXiv:math/0504272 [math.NT], 2005.
James A. Sellers and Roberto Tauraso, Arithmetic properties of MacMahon-type sums of divisors: the odd case, arXiv:2505.12813 [math.NT], 2025.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113414, A113446, A113652 all satisfy A(4n+1)=a(n).
Cf. A004020, A005883, A104794, A052343, A199015 (partial sums).
Number of ways of writing n as a sum of k triangular numbers, for k=1,...: A010054, A008441, A008443, A008438, A008439, A008440, A226252, A007331, A226253, A226254, A226255, A014787, A014809.
Cf. A002175, A053692, A113407, A121444.
Cf. A000217, A001938, A004018.
Cf. A274621 (reciprocal series).
Cf. A001826, A001842.
Cf. A019669, A079006.

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

A := Basis( ModularForms( Gamma1(8), 1), 420); A[2]; /* Michael Somos, Jan 31 2015 */

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:
A002654 := proc(n) sigmamr(n,4,1)-sigmamr(n,4,3) ; end proc:
A008441 := proc(n) A002654(4*n+1) ; end proc:
seq(A008441(n),n=0..90) ; # R. J. Mathar, Mar 23 2011

Plus@@((-1)^(1/2 (Divisors[4#+1]-1)))& /@ Range[0, 104] (* Ant King, Dec 02 2010 *)
a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 2, 0, q] EllipticTheta[ 3, 0, q], {q, 0, n + 1/4}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q]^2, {q, 0, 2 n + 1/2}]; (* Michael Somos, Jun 19 2012 *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 4 n + 1, (-1)^Quotient[#, 2] &]];  (* Michael Somos, Jun 08 2014 *)
QP = QPochhammer; s = QP[q^2]^4/QP[q]^2 + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
TriangleQ[n_] := IntegerQ@Sqrt[8n +1]; Table[Count[FrobeniusSolve[{1, 1}, n], {?TriangleQ}], {n, 0, 104}] (* Robert G. Wilson v, Apr 15 2017 *)

{a(n) = if( n<1, n==0, polcoeff( sum(k=0, (sqrtint(8*n + 1) - 1)\2, x^(k * (k+1)/2), x * O(x^n))^2, n) )};

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, (-1)^(d\2)))}; /* Michael Somos, Sep 02 2005 */

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

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 14 2005 */

{ my(q='q+O('q^166)); Vec(eta(q^2)^4 / eta(q)^2) } \\ Joerg Arndt, Apr 16 2017

ModularForms( Gamma1(8), 1, prec=420).1; # Michael Somos, Jun 08 2014

This sequence is the quadrisection of many sequences. Here are two examples:
a(n) = A002654(4n+1), the difference between the number of divisors of 4*n+1 of form 4*k+1 and the number of form 4*k-1. - David Broadhurst, Oct 20 2002
a(n) = b(4*n + 1), where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Sep 14 2005
G.f.: (Sum_{k>=0} x^((k^2 + k)/2))^2 = (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k in Z} x^(k^2)).
Expansion of Jacobi theta (theta_2(0, sqrt(q)))^2 / (4 * q^(1/4)).
Sum[d|(4n+1), (-1)^((d-1)/2) ].
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = v^3 + 4 * v * w^2 - u^2 * w. - Michael Somos, Sep 14 2005
Given g.f. A(x), then B(q) = q * A(q^4) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = u1 * u3 - (u2 - u6) * (u2 + 3*u6). - Michael Somos, Sep 14 2005
Expansion of Jacobi k/(4*q^(1/2)) * (2/Pi)* K(k) in powers of q^2. - Michael Somos, Sep 14 2005. Convolution of A001938 and A004018. This appears in the denominator of the Jacobi sn and cn formula given in the Abramowitz-Stegun reference, p. 575, 16.23.1 and 16.23.2, where m=k^2. - Wolfdieter Lang, Jul 05 2016
G.f.: Sum_{k>=0} a(k) * x^(2*k) = Sum_{k>=0} x^k / (1 + x^(2*k + 1)).
G.f.: Sum_{k in Z} x^k / (1 - x^(4*k + 1)). - Michael Somos, Nov 03 2005
Expansion of psi(x)^2 = phi(x) * psi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.
Moebius transform is period 8 sequence [ 1, -1, -1, 0, 1, 1, -1, 0, ...]. - Michael Somos, Jan 25 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 1/2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A104794.
Euler transform of period 2 sequence [ 2, -2, ...].
G.f.: q^(-1/4) * eta(q^2)^4 / eta(q)^2. See also the Fine reference.
a(n) = Sum_{k=0..n} A010054(k)*A010054(n-k). - Reinhard Zumkeller, Nov 03 2009
A004020(n) = 2 * a(n). A005883(n) = 4 * a(n).
Convolution square of A010054.
G.f.: Product_{k>0} (1 - x^(2*k))^2 / (1 - x^(2*k-1))^2.
a(2*n) = A113407(n). a(2*n + 1) = A053692(n). a(3*n) = A002175(n). a(3*n + 1) = 2 * A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - Michael Somos, Jun 08 2014
G.f.: exp( Sum_{n>=1} 2*(x^n/n) / (1 + x^n) ). - Paul D. Hanna, Mar 01 2016
a(n) = A001826(2+8*n) - A001842(2+8*n), the difference between the number of divisors 1 (mod 4) and 3 (mod 4) of 2+8*n. See the Ono et al. link, Corollary 1, or directly the Niven et al. reference, p. 165, Corollary (3.23). - Wolfdieter Lang, Jan 11 2017
Expansion of continued fraction 1 / (1 - x^1 + x^1*(1 - x^1)^2 / (1 - x^3 + x^2*(1 - x^2)^2 / (1 - x^5 + x^3*(1 - x^3)^2 / ...))) in powers of x^2. - Michael Somos, Apr 20 2017
Given g.f. A(x), and B(x) is the g.f. for A079006, then B(x) = A(x^2) / A(x) and B(x) * B(x^2) * B(x^4) * ... = 1 / A(x). - Michael Somos, Apr 20 2017
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A002129(k)*a(n-k) for n > 0. - Seiichi Manyama, May 06 2017
From Paul D. Hanna, Aug 10 2019: (Start)
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) - x^(2*k))^(n-k) = Sum_{n>=0} a(n)*x^(2*n).
G.f.: Sum_{n>=0} x^n * Sum_{k=0..n} binomial(n,k) * (x^(2*n+1) + x^(2*k))^(n-k) * (-1)^k = Sum_{n>=0} a(n)*x^(2*n). (End)
From Peter Bala, Jan 05 2021: (Start)
G.f.: Sum_{n = -oo..oo} x^(4*n^2+2*n) * (1 + x^(4*n+1))/(1 - x^(4*n+1)). See Agarwal, p. 285, equation 6.20 with i = j = 1 and mu = 4.
For prime p of the form 4*k + 3, a(n*p^2 + (p^2 - 1)/4) = a(n).
If n > 0 and p are coprime then a(n*p + (p^2 - 1)/4) = 0. The proofs are similar to those given for the corresponding results for A115110. Cf. A000729.
For prime p of the form 4*k + 1 and for n not congruent to (p - 1)/4 (mod p) we have a(n*p^2 + (p^2 - 1)/4) = 3*a(n) (since b(n), where b(4*n+1) = a(n), is multiplicative). (End)
From Peter Bala, Mar 22 2021: (Start)
G.f. A(q) satisfies:
A(q^2) = Sum_{n = -oo..oo} q^n/(1 - q^(4*n+2)) (set z = q, alpha = q^2, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^(2*n)/(1 - q^(4*n+1)) (set z = q^2, alpha = q, mu = 4 in Agarwal, equation 6.15).
A(q^2) = Sum_{n = -oo..oo} q^n/(1 + q^(2*n+1))^2 = Sum_{n = -oo..oo} q^(3*n+1)/(1 + q^(2*n+1))^2. (End)
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) + 2 * Sum_{n>=1, k>=0} (-1)^k * q^(k*(k+2*n+1)+n). - Mamuka Jibladze, May 17 2021
G.f.: Sum_{k>=0} a(k) * q^k = Sum_{k>=0} (-1)^k * q^(k*(k+1)) * (1 + q^(2*k+1))/(1 - q^(2*k+1)). - Mamuka Jibladze, Jun 06 2021
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Oct 15 2022

More terms and information from Michael Somos, Mar 23 2003

A122856 Expansion of f(x, x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 14 2006

nonn

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

			G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + 2*x^13 + ...
G.f. = q^2 + 2*q^5 + q^8 + 2*q^17 + 2*q^20 + 2*q^26 + 2*q^29 + q^32 + ...

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. A002654, A035154, A089801, A113446, A121444, A122865, A258278.

a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 2}, Sum[ KroneckerSymbol[ -4, d], {d, Divisors@m}]]]; (* Michael Somos, Nov 14 2011 *)
QP = QPochhammer; s = (QP[q^2]^2*QP[q^3]*(QP[q^12]/(QP[q]*QP[q^4]*QP[q^6]) ))^2 + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^(1/3)] - EllipticTheta[ 3, 0, x^3])^2 / (4 x^(2/3)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)])^2 / (2 x^(3/4)), {x, 0, n}]; (* Michael Somos, Jan 19 2017 *)

{a(n) = if( n<0, 0, n = 3*n+2; sumdiv(n, d, (d%4==1) - (d%4==3)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^2, n))};

Expansion of (chi(x) * psi(-x^3))^2 in powers of x where chi(), psi() are Ramanujan theta functions.
Expansion of q^(-2/3) * (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^2 in powers of q.
Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, -2, ...].
a(4*n + 3) = a(8*n + 4) = 0. a(n) = A002654(3*n + 2) = A035154(3*n + 2) = A113446(3*n + 2). a(2*n) = A122865(n). a(4*n + 1) = 2 * A121444(n). a(4*n + 2) = A122856(n).
a(n) = (-1)^n * A258278(n). Convolution square of A089801.

A122865 Expansion of chi(x) * phi(x^3) * psi(-x^3) in powers of x where chi(), phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 15 2006

nonn

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

			G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 3*x^8 + 2*x^11 + 2*x^12 + 2*x^13 + ...
G.f. = q + q^4 + 2*q^10 + 2*q^13 + q^16 + 3*q^25 + 2*q^34 + 2*q^37 + ...

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. A002175, A002654, A035154, A113446, A122856, A122864, A163746, A258277.

phi[q_] := EllipticTheta[3, 0, q]; chi[q_] := ((1 - InverseEllipticNomeQ[q]) * InverseEllipticNomeQ[q]/(16*q))^(-1/24); psi[q_] := (1/2)*q^(-1/8)*EllipticTheta[ 2, 0, q^(1/2)]; s = Series[ chi[q]*phi[q^3]*psi[-q^3], {q, 0, 104}]; a[n_] := Coefficient[s, q, n];
(* or *) a[n_] := If[n == 0, 1, Sum[Boole[Mod[d, 4] == 1] - Boole[Mod[d, 4] == 3], {d, Divisors[3n+1]}]]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Feb 17 2015, after PARI code *)
a[ n_] := If[ n < 0, 0, DivisorSum[ 3 n + 1, KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3] QPochhammer[ x^6], {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^4 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))};

{a(n) = my(A, p, e); if(n <0, 0, n = 3*n+1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, -2*(-1)^e, p%4==1, e+1, 1-e%2)))};

{a(n) = if( n<0, 0, n = 3*n+1; sumdiv(n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Apr 19 2007 */

Expansion of chi(x) * f(x^3) * f(-x^6) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 02 2015
Expansion of q^(-1/3) * eta(q^2)^2 * et(q^6)^4 / (eta(q) * eta(q^3) * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [1, -1, 2, 0, 1, -4, 1, 0, 2, -1, 1, -2, ...]. - Michael Somos, Apr 19 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258228. - Michael Somos, Sep 02 2015
a(n) = A002654(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122864(3*n + 1) = A163746(3*n + 1).
a(n) = (-1)^n * A258277(n). a(2*n + 1) = A122856(n). - Michael Somos, Sep 02 2015
a(4*n) = A002175(n). a(4*n + 2) = 0. - Michael Somos, Jan 19 2017

A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.

Original entry on oeis.org

1, 2, 3, 2, 1, 2, 2, 4, 2, 2, 1, 0, 4, 2, 3, 2, 2, 4, 0, 2, 2, 0, 4, 2, 3, 0, 2, 6, 2, 2, 1, 2, 0, 2, 2, 2, 2, 4, 2, 0, 4, 4, 4, 0, 1, 2, 0, 4, 2, 0, 2, 2, 5, 2, 0, 2, 2, 4, 4, 2, 0, 2, 4, 2, 2, 0, 4, 0, 0, 2, 3, 2, 4, 2, 0, 4, 0, 6, 2, 4, 1, 0, 4, 2, 2, 2, 2, 0, 0, 2, 0, 2, 8, 2, 2, 0, 2, 4, 0, 4, 2, 2, 3, 2, 2

Offset: 0

N. J. A. Sloane

nonn

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

Number of ways to write n as an ordered sum of 2 generalized pentagonal numbers. - Ilya Gutkovskiy, Aug 14 2017

			G.f. = 1 + 2*x + 3*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 4*x^7 + 2*x^8 + 2*x^9 + ...
G.f. = q + 2*q^13 + 3*q^25 + 2*q^37 + q^49 + 2*q^61 + 2*q^73 + 4*q^85 + 2*q^97 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

John Cerkan, Table of n, a(n) for n = 0..10000
J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002654, A008441, A008442, A035154, A035181, A035184, A112301, A113406.
Cf. A113652, A116604, A121363, A121450, A122856, A122864, A122865, A125061.
Cf. A125079, A129447, A129448, A132004, A134013, A134015, A138741, A138746.
Cf. A138950, A138952, A163746, A258210, A258228, A258256, A258279, A258292.

series(mul( ( (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2, n = 1..100), q, 101):
seq(coeftayl(%, q = 0, n), n = 0..100); # Peter Bala, Jan 05 2025

ed[n_]:=Module[{divs=Divisors[12n+1]},Count[divs,?(Mod[#,4] == 1&)]- Count[divs,?(Mod[#,4]==3&)]]; Array[ed,110,0] (* Harvey P. Dale, Jul 01 2012 *)
a[ n_] := If[ n < 0, 0, With[ {m = 12 n + 1}, Sum[ KroneckerSymbol[ 4, d], {d, Divisors[m]}]]]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3]^2 / (QPochhammer[ x] QPochhammer[ x^6]))^2, {x, 0, n}]; (* Michael Somos, Apr 23 2014 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] / QPochhammer[ x, x^2])^2, {x, 0, n}]; (* Michael Somos, May 25 2015 *)

{a(n) = if( n<0, 0, n = 12*n + 1; sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Sep 19 2005 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A) * eta(x^6 + A)))^2, n))}; /* Michael Somos, Jun 02 2012 */

Expansion of (phi(-x^3) / chi(-x))^2 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/12) * (eta(q^2) * eta(q^3)^2 / (eta(q) * eta(q^6)))^2 in powers of q. - Michael Somos, Sep 19 2005
Euler transform of period 6 sequence [ 2, 0, -2, 0, 2, -2, ...]. - Michael Somos, Sep 19 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258279. - Michael Somos, May 25 2015
From Michael Somos, Jun 02 2012: (Start)
a(n) = A008441(3*n) = A121363(3*n) = A122865(4*n) = A122856(8*n).
a(n) = A116604(6*n) = A125079(6*n) = A129447(6*n) = A138741(6*n).
a(n) = A(12*n+1) where A = A002654, A008442, A035154, A035181, A035184, A112301, A113406, A113652, A121450, A122864, A125061, A129448, A132004, A134013, A134015, A138746, A138950, A138952, A163746.  (End)
From Michael Somos, May 25 2015: (Start)
a(n) = A258277(4*n) = A258278(8*n) = A258291(3*n).
a(n) = - A258210(12*n + 1) = A258228(12*n + 1) = A258256(12*n + 1).
2*a(n) = A258279(12*n + 1) = - A258292(12*n + 1). (End)
G.f.: (Sum_{k = -oo..oo} x^(k*(3*k-1)/2))^2. - Ilya Gutkovskiy, Aug 14 2017
G.f.: ( Product_{n >= 1} (1 + q^n)*(1 - q^(3*n))/(1 + q^(3*n)) )^2. - Peter Bala, Jan 05 2025

A113446 Expansion of (phi(q)^2 - phi(q^3)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Nov 02 2005

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

			G.f. = q + q^2 - q^3 + q^4 + 2*q^5 - q^6 + q^8 + q^9 + 2*q^10 - q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A008441, A019673, A035154, A138949.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] + A[3] - A[4] + A[5]; /* Michael Somos, Jan 31 2015 */

a[ n_] := If[ n < 1, 0, (-1)^IntegerExponent[ n, 3] Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) EllipticTheta[ 2, 0, q^(3/2)]^3 / EllipticTheta[ 2, 0, q^(1/2)] (EllipticTheta[ 3, 0, q] / EllipticTheta[ 3, 0, q^3]), {q, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = if( n<1, 0, (-1)^valuation(n, 3) * sumdiv(n, d, kronecker(-36, d)))};

{a(n) = if( n<1, 0, direuler(p=2, n, if( p==3, 1 / (1 + X), 1 / (1 - X) / (1 - kronecker(-36, p) * X)))[n])};

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^6 + A) * eta(x^12 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2), n))};

Expansion of (eta(q^2)^3 * eta(q^6) * eta(q^12)^2) / (eta(q) * eta(q^3) * eta(q^4)^2) in powers of q.
Euler transform of period 12 sequence [1, -2, 2, 0, 1, -2, 1, 0, 2, -2, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, -2, 0, 1, 0, -1, 0, 2, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = 1, a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f.: ((Sum_{k} x^(k^2))^2 - (Sum_{k} x^(3*k^2))^2) / 4.
G.f.: Sum_{k>0} x^(3*k-1) / (1 + x^(6*k-2)) + x^(3*k-2)/(1 + x^(6*k-4)).
G.f.: Sum_{k>0} x^k * (1 - x^(2*k))^2 / (1 + x^(6*k)).
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^k)^3 * (1 + x^(3*k)) * (1 + x^(4*k) + x^(8*k))^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138949.
a(n) = (-1)^e * A035154(n) where 3^e is the highest power of 3 dividing n.
a(4*n + 1) = A008441(n).
Expansion of q * f(-q, -q^5) * f(q, q^5)^2 / phi(-q^3) in powers of q where phi(), f(,) are Ramanujan theta functions. - Michael Somos, Jan 31 2015
Expansion of q * (psi(q^3)^3 / psi(q)) * (phi(q) / phi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/6 (A019673). - Amiram Eldar, Nov 24 2023

A125079 Excess of number of divisors of 2n+1 of form 12k+1, 12k+5 over those of form 12k+7, 12k+11.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 18 2006

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

			1 + x + 2*x^2 + x^4 + 2*x^6 + 2*x^7 + 2*x^8 + 3*x^12 + x^13 + 2*x^14 + ...
q + q^3 + 2*q^5 + q^9 + 2*q^13 + 2*q^15 + 2*q^17 + 3*q^25 + q^27 + 2*q^29 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 82, Eq. (32.56).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A008441, A019670, A035154, A121444.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, With[{m = 2 n + 1}, Sum[ KroneckerSymbol[ -36, d], { d, Divisors[ m]}]]]

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -36, d)))}

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( 6, d) * (-1)^(d\12)))}

{a(n) = if( n<0, 0, if( n%6==1, n\=3, 1); sumdiv( 2*n + 1, d, kronecker( -4, d)) )}

{a(n) = local(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 0, if( p==3, 1, if( p%4==1, e+1, !(e%2)))))))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^6 + A)^2), n))}

a008441(n) = sumdiv(4*n+1, d, (-1)^(d\2));
a(n) = if(n%2==0, a008441(n/2), if(n%6==1, a008441((n-1)/6))); \\ Seiichi Manyama, Oct 11 2024

Expansion of q^(-1/2) * eta(q^3)^3 *eta(q^4) * eta(q^12) / (eta(q) * eta(q^6)^2) in powers of q.
Expansion of phi(-q^3) * psi(-q^3) / (chi(-q) * chi(-q^2)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, 0, 1, 0, -2, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 4), b(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138745.
a(6*n + 3) = a(6*n + 5) = 0. a(6*n) = A002175(n). a(6*n + 1) = a(2*n) = A008441(n). a(6*n + 2) = 2 * A121444(n). a(n) = A035154(2*n + 1).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Dec 28 2023
a(6*n + 3) = a(6*n + 5) = 0. a(6*n + 1) = A008441(n). a(6*n + k) = A008441(3*n + k/2) for k=0,2,4. - Seiichi Manyama Oct 11 2024

A122857 Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 0, 2, 2, 4, 0, 2, 4, 0, 4, 2, 4, 2, 0, 4, 0, 0, 0, 2, 6, 4, 2, 0, 4, 4, 0, 2, 0, 4, 0, 2, 4, 0, 4, 4, 4, 0, 0, 0, 4, 0, 0, 2, 2, 6, 4, 4, 4, 2, 0, 0, 0, 4, 0, 4, 4, 0, 0, 2, 8, 0, 0, 4, 0, 0, 0, 2, 4, 4, 6, 0, 0, 4, 0, 4, 2, 4, 0, 0, 8, 0, 4, 0, 4, 4, 0, 0, 0, 0, 0, 2, 4, 2, 0, 6, 4, 4, 0, 4

Offset: 0

Michael Somos, Sep 14 2006

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

			G.f. = 1 + 2*q + 2*q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 2*q^6 + 2*q^8 + 2*q^9 + 4*q^10 + ...

B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 197, Entry 44.

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. A002654, A008441, A019693, A035154, A113446, A125079, A125061, A132003.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Jul 09 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 2, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)

{a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -36, d)))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 1, p%12<6, e+1, !(e%2) )))};

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],
     [p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

Expansion of eta(q^2)^3 * eta(q^3)^2 * eta(q^6) / (eta(q)^2 * eta(q^4)* eta(q^12)) in powers of q.
Expansion of 2 * psi(q) * psi(q^2) * psi(q^3) / psi(q^6) - phi(q^3)^2 in powers of q. - Michael Somos, Jul 09 2013
Euler transform of period 12 sequence [ 2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -2, ...].
Moebius transform is period 12 sequence [ 2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, ...].
a(12*n + 7) = a(12*n + 11) = 0.
a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = b(3^e) = 1, b(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) == (1-(-1)^e)/2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A125061.
A035154(n) = a(n) / 2 if n > 0. A008441(n) = a(4*n + 1) / 2. A125079(n) = a(2*n + 1) / 2. A113446(3*n + 1) = A002654(3*n + 1) = a(3*n + 1) / 2.
a(n) = (-1)^n * A132003(n). Expansion of (phi(-q^3) / phi(-q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 05 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.0943951... (A019693). - Amiram Eldar, Nov 21 2023

A122864 Expansion of eta(q^3)^2 * eta(q^4) * eta(q^6)^2 * eta(q^36) / (eta(q) * eta(q^9) * eta(q^12)^2) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 15 2006

			q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + q^8 - 2*q^9 + 2*q^10 + 2*q^12 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A002564, A019670, A035154, A113446, A122856, A122865.

eta[x_] := x^(1/24)*QPochhammer[x]; A122864[n_] := SeriesCoefficient[ (eta[q^3]^2*eta[q^4]*eta[q^6]^2*eta[q^36])/(eta[q]*eta[q^9]*eta[q^12]^2), {q, 0, n}]; Table[A122864[n], {n, 50}] (* G. C. Greubel, Sep 16 2017 *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^36 + A) /(eta(x + A) * eta(x^9 + A) * eta(x^12 + A)^2), n))}

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p==3, -2*(-1)^e, if( p%4==1, e+1, 1-e%2))))))}

Euler transform of period 36 sequence [ 1, 1, -1, 0, 1, -3, 1, 0, 0, 1, 1, -2, 1, 1, -1, 0, 1, -2, 1, 0, -1, 1, 1, -2, 1, 1, 0, 0, 1, -3, 1, 0, -1, 1, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, 1, 0, 1, 0, -1, 0, -4, 0, -1, 0, 1, 0, 1, 0, 1, 0, -1, 0, -1, 0, -1, 0, 1, 0, 4, 0, 1, 0, -1, 0, -1, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = 1, a(3^e) = 2*(-1)^(e+1) if e>0, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
a(3*n) = 2 * A113446(n). a(3*n + 1) = A002564(3*n + 1) = A035154(3*n + 1) = A113446(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Oct 15 2022

A281451 Expansion of x * f(x, x) * f(x, x^17) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 23 2017

nonn

			G.f. = x + 3*x^2 + 2*x^3 + 2*x^5 + 2*x^6 + 2*x^10 + 2*x^11 + 2*x^17 + ...
G.f. = q^16 + 3*q^25 + 2*q^34 + 2*q^52 + 2*q^61 + 2*q^97 + 2*q^106 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002654, A004018, A019670, A035154, A105673, A113446, A122856, A122857, A122864, A122865.
Cf. A125061, A129448, A138949, A138950, A163746, A256269, A256276, A256280, A258034.
Cf. A258228, A258256, A258278, A258292, A259761.
Cf. A281452, A281453, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 7, KroneckerSymbol[ -4, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 7])];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x, x^18] QPochhammer[ -x^17, x^18] QPochhammer[ x^18], {x, 0, n}];

{a(n) = if( n<0, 0, sumdiv(9*n + 7, d, (d%4==1) - (d%4==3)))};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 7); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%4==1, e+1, 1-e%2)))};

{a(n) = if( n<0, 0, my(m = 9*n + 7, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 4 || k%9 == 5), s+=(j>0)+1)); s)};

f(x,x^m) = 1 + Sum_{k>=1} x^((m+1)*k*(k-1)/2) (x^k + x^(m*k)). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 8*k)).
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-17)) * (1 + x^(18*k-1)) * (1 - x^(18*k)).
a(4*n) = a(8*n + 7) = a(16*n + 13) = a(32*n + 9) = a(49*n + 7) = a(98*n + 14) = 0.
a(4*n + 1) = A281452(n).  a(8*n + 3) = 2 * A281491(n).  A(16*n + 1) = A281453(n).
a(32*n + 25) = 2 * A281490(n). a(64*n + 49) = a(n). a(128*n + 17) = 2 * A281492(n).
a(n) = A122865(3*n + 2). a(n) = A122856(6*n + 4) = A258278(6*n + 4).
a(n) = b(9*n + 7) where b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 7) where b = A105673, A122857, A258034, A259761.  -2 * a(n) = b(9*n + 7) where b = A138949, A256280, A258292.
a(n) = - A256269(9*n + 7). 4 * a(n) = A004018(9*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

A121450 Expansion of (theta_4(q^3)^2 - theta_4(q)^2)/4 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jul 30 2006

			G.f. = x - x^2 - x^3 - x^4 + 2*x^5 + x^6 - x^8 + x^9 - 2*x^10 + x^12 + ... - _Michael Somos_, Sep 07 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000

A035154(n) = |a(n)|.

Apart from signs, same as A035154 and A113446.

eta[q_]:= q^(1/24)*QPochhammer[q]; Rest[CoefficientList[Series[eta[q]* eta[q^3]*eta[q^12]^3/(eta[q^4]*eta[q^6]^2), {q, 0, 100}], q]] (* G. C. Greubel, Apr 19 2018 *)

{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==2, -1, p==3, (-1)^e, p%4==1, (e+1), (1+(-1)^e)/2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^3 / (eta(x^4 + A) * eta(x^6 + A)^2), n))};

Expansion of eta(q) * eta(q^3) * eta(q^12)^3 /(eta(q^4) * eta(q^6)^2) in powers of q.
Euler transform of period 12 sequence [-1, -1, -2, 0, -1, 0, -1, 0, -2, -1, -1, -2, ...].
Moebius transform is period 24 sequence [1, -2, -2, 0, 1, 4, -1, 0, 2, -2, -1, 0, 1, 2, -2, 0, 1, -4, -1, 0, 2, 2, -1, 0, ...].
Multiplicative with a(2^e) = -1 if e > 0, a(3^e) = (-1)^e, a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k)/ (1 + x^(3*k)) = x * Product_{k>0} (1 - x^k)^2 * (1 - x^k + x^(2*k)) * (1 + x^(2*k))^2 *(1 + x^k + x^(2*k))^2 * (1 - x^(2*k) + x^(4*k))^3.

cp's OEIS Frontend

A008441 Number of ways of writing n as the sum of 2 triangular numbers.

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A122856 Expansion of f(x, x^5)^2 in powers of x where f(, ) is Ramanujan's general theta function.

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A122865 Expansion of chi(x) * phi(x^3) * psi(-x^3) in powers of x where chi(), phi(), psi() are Ramanujan theta functions.

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A002175 Excess of number of divisors of 12n+1 of form 4k+1 over those of form 4k+3.

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A113446 Expansion of (phi(q)^2 - phi(q^3)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

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