A053692 -id:A053692 - cp's OEIS frontend

A294065 Row 2 in rectangular array A292929.

2, -4, 8, -14, 18, -36, 56, -74, 116, -164, 224, -324, 442, -592, 808, -1074, 1410, -1860, 2416, -3102, 4010, -5112, 6464, -8204, 10294, -12860, 16072, -19914, 24586, -30356, 37248, -45534, 55608, -67604, 81928, -99182, 119608, -143832, 172760, -206834, 247048, -294676, 350504, -416080, 493248, -583340, 688616, -811740, 954974, -1121564, 1315504, -1540210, 1800434, -2102060, 2450224, -2852040

Offset: 0

Paul D. Hanna, Oct 23 2017

sign

			G.f.: A(q) = 2 - 4*q + 8*q^2 - 14*q^3 + 18*q^4 - 36*q^5 + 56*q^6 - 74*q^7 + 116*q^8 - 164*q^9 + 224*q^10 - 324*q^11 + 442*q^12 - 592*q^13 + 808*q^14 - 1074*q^15 + 1410*q^16 - 1860*q^17 + 2416*q^18 - 3102*q^19 + 4010*q^20 +...
RELATED SERIES.
Let R1(q) denote the g.f. of row 1 (with offset 0) in array A292929, then
A(q)/R1(q) = 1 + q^2 + q^3 - 3*q^5 + q^6 + 4*q^7 + q^8 - 3*q^9 + q^10 + 3*q^11 + q^12 - 5*q^13 + q^14 + 7*q^15 - 11*q^17 + 16*q^19 + 2*q^20 - 18*q^21 + 21*q^23 + q^24 - 27*q^25 + q^26 + 38*q^27 + q^28 - 55*q^29 + 2*q^30 +...
then it appears that the even bisection of A(q)/R1(q) forms a g.f. of A053692:
(A(q)/R1(q) + A(-q)/R1(-q))/2 = Product_{n>=1} (1 - q^(16*n))^2*(1 + q^(4*n-2)).

Paul D. Hanna, Table of n, a(n) for n = 0..380

Cf. A292929, A293132, A294066, A294067.

nmax = 55; kmax = Ceiling[Sqrt[nmax]];
Q[q_] := Sum[(x - q^k)^k, {k, -kmax, kmax}];
S[q_] := Sqrt[Q[q]/Q[-q]];
row[n_] := (1/q^n)*SeriesCoefficient[Sqrt[Q[q]/Q[-q]], {x, 0, n}] + O[q]^nmax // CoefficientList[#, q]&;
row[2] (* Jean-François Alcover, Nov 04 2017 *)

A112301 Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 02 2005, Oct 02 2007

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

			G.f. = q - 2*q^2 + 2*q^5 + q^9 - 4*q^10 + 2*q^13 + 2*q^17 - 2*q^18 + 3*q^25 - ...

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. A008441, A053692, A113407, A134013.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 2, 0, q^4] / 2, {q, 0, n}]; (* Michael Somos, Oct 19 2013 *)
QP = QPochhammer; s = (QP[q]*QP[q^16])^2/(QP[q^2]*QP[q^8]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^16 + A))^2 / (eta(x^2 + A) * eta(x^8 + A)), n))};

{a(n) = if( n>0 & (n+1)%4\2, (n%2*3 - 2) * sumdiv( n / gcd(n, 2), d, (-1)^(d\2)))};

Expansion of q * phi(-q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (phi(-q^2)^2 - phi(-q)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 16 sequence [ -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, ...].
a(n) is ultiplicative with a(2) = -2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
Moebius transform is period 16 sequence [ 1, -3, -1, 2, 1, 3, -1, 0, 1, -3, -1, -2, 1, 3, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 4 (t/i) f(t) where q = exp(2 Pi i t).
G.f.: x * Product_{k>0} (1 - x^k)^2 * (1 + x^(8*k))^2 * (1 + x^(2*k)) * (1 + x^(4*k)).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = -2 * A008441(n).
a(n) = -(-1)^n * A134013(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n).

A134013 Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 02 2007

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

			G.f. = q + 2*q^2 + 2*q^5 + q^9 + 4*q^10 + 2*q^13 + 2*q^17 + 2*q^18 + 3*q^25 + ...

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. A003881, A008441, A053692, A112301, A113407, A134014.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^4], {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)

{a(n) = if( n>0 && (n+1)%4\2, (n%4) * sumdiv( n/gcd(n,2), d, (-1)^(d\2)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^8 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^16)^2 / ( eta(q)^2 * eta(q^4)^2 * eta(q^8) ) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, ...].
Moebius transform is period 16 sequence [ 1, 1, -1, -2, 1, -1, -1, 0, 1, 1, -1, 2, 1, -1, -1, 0, ...].
a(n) is multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e)/2 if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134014.
a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = 2 * a(4*n + 1).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A112301(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n = 5) = 2 * A053692(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 (A003881). - Amiram Eldar, Nov 24 2023

A132004 Expansion of (1 - phi(q^3) / phi(q) * phi(-q^2) * phi(-q^6)) / 2 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

Offset: 1

Michael Somos, Aug 06 2007

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

			G.f. = x - x^2 + x^3 - x^4 + 2*x^5 - x^6 - x^8 + x^9 - 2*x^10 - x^12 + 2*x^13 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

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. A008441, A035154, A053692, A122856, A122865, A125079, A132003, A258277, A258278.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Nov 01 2015 *)

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3)) / 2, n))};

{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, -1, p%4==1, e+1, 1-e%2)))};

Expansion of (1 - eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3)) / 2 in powers of q.
a(n) is multiplicative with a(2^e) = 2*0^e - 1, a(3^e) = 1, 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>0} x^k / (1 + x^k) * Kronecker(-36, k).
a(3*n) = a(n). -2 * a(n) = A132003(n) unless n = 0. a(2*n) = - A035154(n). a(2*n + 1) = A125079(n).
a(n) = (-1)^n * A035154(n). a(12*n + 7) = a(12*n + 11) = 0. - Michael Somos, Nov 01 2015
a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 2) = - A125079(n). - Michael Somos, Nov 01 2015
a(6*n) = - A035154(n). a(6*n + 2) = - A122865(n). a(6*n + 4) = - A122856(n). - Michael Somos, Nov 01 2015
a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). - Michael Somos, Nov 01 2015

A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 3, 2, 3, 4, 2, 0, 3, 5, 4, 2, 2, 4, 0, 0, 3, 6, 5, 2, 4, 0, 2, 0, 2, 7, 4, 4, 0, 4, 0, 0, 3, 4, 6, 0, 5, 4, 2, 0, 4, 6, 0, 2, 2, 4, 0, 0, 2, 3, 7, 4, 4, 4, 4, 0, 0, 4, 4, 2, 0, 4, 0, 0, 3, 8, 4, 2, 6, 0, 0, 0, 5, 6, 4, 2, 2, 0, 0, 0, 4, 7, 6, 2, 0, 8, 2

Offset: 0

Michael Somos, Jun 29 2014

nonn

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

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

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A093709, A107635, A121373, A244526, A244543.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 3*A[2] + 3*A[3];

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {3, 0, -1, 0, 1, 0, -3, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 3, 0, -1, 0, 1, 0, -3][d%8 + 1]))};

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

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 3*A[1] + 3*A[2];

Expansion of f(-q^3, -q^5)^2 * phi(q) / psi(-q) = f(-q^3, -q^5)^2 * chi(q)^3 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [3, -3, 1, 0, 1, -3, 3, -2, ...].
Moebius transform is period 8 sequence [3, 0, -1, 0, 1, 0, -3, 0, ...].
Convolution product of A244526 and A107635. Convolution product of A000122 and A093709.
a(n) = (A004018(n) + A033715(n)) / 2 = A244543(2*n).
a(2*n) = a(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 4*A053692(n). a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 + 1/sqrt(2))/2 = 2.681517... . - Amiram Eldar, Jun 08 2025

A226194 Expansion of f(-x^1, -x^7) * f(-x^3, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, May 30 2013

sign

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

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^10 + x^12 - x^13 + x^14 - 2*x^15 + ...
G.f. = q^5 - q^13 - q^29 + q^37 - q^45 + q^53 - q^61 + 2*q^85 + q^101 - q^109 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A053692, A226192.

a[ n_] := SeriesCoefficient[ QPochhammer[ q^8]^2 / QPochhammer[ -q, q], {q, 0, n}];

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

{a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 5; A = factor(n); simplify( -I/2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p%4 == 3, if( e%2, 0, (-1)^(e * (p+1) / 8)), (e+1) * I^(e * (p-1) / 4)))))};

Expansion of q^(-5/8) * eta(q) * eta(q^8)^2 / eta(q^2) in powers of q.
Euler transform of period 8 sequence [-1, 0, -1, 0, -1, 0, -1, -2, ...].
a(n) = -I/2 * b(8*n + 5) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (-1)^(e * (p+1)/8) * (1 + (-1)^e) / 2 if p == 3 (mod 4), b(p^e) = (e+1) * I^(e * (p-1)/4) if p == 1 (mod 4).
G.f.: Product_{k>0} (1 - x^(8*k))^2 / (1 + x^k).
a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = -a(n).
a(n) = (-1)^n * A053692(n).

A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, -2, 1, 4, -2, 0, 1, -1, 4, -2, -2, 4, 0, 0, 1, 2, -1, -2, 4, 0, -2, 0, -2, 5, 4, -4, 0, 4, 0, 0, 1, -4, 2, 0, -1, 4, -2, 0, 4, 2, 0, -2, -2, 4, 0, 0, -2, 1, 5, -4, 4, 4, -4, 0, 0, -4, 4, -2, 0, 4, 0, 0, 1, 8, -4, -2, 2, 0, 0, 0, -1, 2, 4, -2, -2, 0, 0

Offset: 1

Michael Somos, Jun 30 2014

sign

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

			G.f. = q + q^2 - 2*q^3 + q^4 + 4*q^5 - 2*q^6 + q^8 - q^9 + 4*q^10 - 2*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A107635, A121373, A143259, A243747, A244560.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[2] + A[3];

a[ n_] := If[ n < 1, 0, Sum[ {1, 0, -3, 0, 3, 0, -1, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] - EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, 0, sumdiv(n, d, [0, 1, 0, -3, 0, 3, 0, -1][d%8 + 1]))};

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

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[1] + A[2];

Expansion of q * f(-q, -q^7)^2 * phi(q) / psi(-q) = q * f(-q, -q^7)^2 * chi(q)^3 in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [1, -3, 3, 0, 3, -3, 1, -2, ...].
Moebius transform is period 8 sequence [1, 0, -3, 0, 3, 0, -1, 0, ...].
Convolution product of A244560 and A107635. Convolution product of A000122 and A143259.
a(n) = (A004018(n) - A033715(n)) / 2 = A243747(2*n).
a(2*n) = a(n). a(8*n + 3) = -2 * A033761(n). a(8*n + 5) = 4 * A053692(n). a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 - 1/sqrt(2))/2 = 0.460075... . - Amiram Eldar, Jun 08 2025

A053693 Number of self-conjugate 8-core partitions of n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 1, 1, 5, 2, 3, 4, 4, 5, 3, 4, 4, 6, 4, 5, 6, 4, 5, 7, 6, 7, 7, 5, 7, 7, 6, 5, 8, 5, 5, 6, 6, 6, 13, 11, 4, 11, 7, 9, 9, 6, 11, 12, 10, 8, 13, 9, 8, 15, 9, 7, 12, 8, 10, 14, 9, 10, 13, 13, 8, 16, 12, 12, 15, 8, 9, 14, 12, 11, 19, 11, 12, 18, 14, 11, 17

Offset: 0

James Sellers, Feb 14 2000

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + 2*x^9 + 2^x*10 + 2*x^11 + ...
G.f. = q^21 + q^29 + q^45 + q^53 + q^61 + q^69 + q^77 + 2*q^85 + 2*q^93 + 2*q^101 + ...

F. Garvan, D. Kim and D. Stanton, Cranks and t-cores, Inventiones Math. 101 (1990) 1-17.

Cf. A053692.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^2 QPochhammer[ x^16]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^16 + A)^4 / (eta(x + A) * eta(x^4 + A)), n))}; /* Michael Somos, Apr 28 2003 */

Euler transform of period 16 sequence [ 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, -4, ...]. - Michael Somos, Apr 28 2003
Expansion of q^(-21/8) * eta(q^2)^2 * eta(q^16)^4 / (eta(q) * eta(q^4)) in powers of q. - Michael Somos, Apr 28 2003
G.f.: product((1-q^(16*i))^4*(1-q^(4*i-2))/(1-q^(2*i-1)), i=1..infinity)

A244544 Expansion of (phi(q) + phi(q^2))^2 / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 2, 0, 3, 4, 4, 2, 2, 2, 0, 0, 3, 4, 5, 2, 4, 0, 2, 0, 2, 4, 4, 4, 0, 2, 0, 0, 3, 4, 6, 0, 5, 2, 2, 0, 4, 4, 0, 2, 2, 2, 0, 0, 2, 2, 7, 4, 4, 2, 4, 0, 0, 4, 4, 2, 0, 2, 0, 0, 3, 4, 4, 2, 6, 0, 0, 0, 5, 4, 4, 2, 2, 0, 0, 0, 4, 6, 6, 2, 0, 4, 2

Offset: 0

Michael Somos, Jun 29 2014

nonn

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

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

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

Cf. A033761, A053692, A093709, A244540.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 2*A[2] + 3*A[3];

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {2, 1, 0, 0, 0, -1, -2, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2])^2 / 4, {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 2, 1, 0, 0, 0, -1, -2][d%8 + 1]))};

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( (A + subst(A, x, x^2))^2 / 4, n))};

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 2*A[1] + 3*A[2];

Expansion of f(-q^3, -q^5)^4 / psi(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [ 2, 0, -2, 2, -2, 0, 2, -2, ...].
Moebius transform is period 8 sequence [ 2, 1, 0, 0, 0, -1, -2, 0, ...].
Convolution square of A093709.
a(2*n) = A244540(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 2*A053692(n). a(8*n + 7) = 0.

A286813 Number of positive odd solutions to equation x^2 + 8y^2 = 8n + 9.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 0, 0, 0, 1, 2, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 2, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 2, 0

Offset: 0

Seiichi Manyama, May 28 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.

Related to the number of positive odd solutions to equation x^2 + k*y^2 = 8*n + k + 1: A008441 (k=1), A033761 (k=2), A033762 (k=3), A053692 (k=4), A033764 (k=5), A259896 (k=6), A035162 (k=7), this sequence (k=8).

Expansion of q^(-9/8) * (eta(q^2) * eta(q^16))^2 / (eta(q) * eta(q^8)) in powers of q.

Euler Transform of -(-2*x^8-x^7-1)/(x^9+x^8+x+1) (o.g.f.). - Simon Plouffe, Jun 23 2018

cp's OEIS Frontend

A294065 Row 2 in rectangular array A292929.

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A112301 Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers of q.

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A134013 Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

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A132004 Expansion of (1 - phi(q^3) / phi(q) * phi(-q^2) * phi(-q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

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A226194 Expansion of f(-x^1, -x^7) * f(-x^3, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

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A244554 Expansion of phi(q) * (phi(q) - phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

A286813 Number of positive odd solutions to equation x^2 + 8y^2 = 8n + 9.