A122856 -id:A122856 - cp's OEIS frontend

A091400 a(n) = Product_{ odd primes p | n } (1 + Legendre(-1,p) ).

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

Offset: 1

N. J. A. Sloane, Mar 02 2004

			G.f. = x + x^2 + x^4 + 2*x^5 + x^8 + 2*x^10 + 2*x^13 + x^16 + 2*x^17 + 2*x^20 + ...

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2) (but without the restriction that a(4k) = 0).

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

Cf. A060294, A091379, A122856, A122865, A129448.

with(numtheory): A091400 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := 1; for i from 1 to nops(t1) do if t1[i][1] > 2 then t2 := t2*(1+legendre(-1,t1[i][1])); fi; od: t2; end;
with(numtheory): seq(mul(1+legendre(-1,p),p in select(isprime, divisors(n) minus {2})),n=1..105); # Peter Luschny, Apr 20 2016

Legendre[-1, p_] := Which[p==2, 0, Mod[p, 4]==1, 1, True, -1]; a[1] = 1; a[n_] := Times @@ (Legendre[-1, #] + 1&) /@ FactorInteger[n][[All, 1]]; Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)
Join[{1},Table[Product[1+JacobiSymbol[-1,p],{p,Complement[FactorInteger[n][[All, 1]], {2}]}], {n,2,105}]] (* Peter Luschny, Apr 20 2016 *)

{a(n)=if(n<1,0,sumdiv(n,d,(-1)^bigomega(d)*moebius(d)*if(d%2,(-1)^(d\2),0)))} \\ Benoit Cloitre, Apr 17 2016

Here we use the definition that Legendre(-1, 2) = 0, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4. This is Shimura's definition, which is different from Maple's.
a(n) is multiplicative with:
   a(2^e) = 1 for e >= 0,
   a(p^e) = 0 if p == 3 (mod 4) for e > 0,
   a(p^e) = 2 if p == 1 (mod 4) for e > 0.
(corrected by Werner Schulte, Dec 12 2020).
a(2*n) = a(n). a(3*n) = a(4*n + 3) = 0.
a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n).
a(n) = Sum_{d|n} b(d)*(-1)^bigomega(d)*moebius(d) where b(2n)=0 and b(2n+1)=(-1)^n. - Benoit Cloitre, Apr 17 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/Pi = 0.636619... (A060294). - Amiram Eldar, Oct 11 2022

Definition clarified by Peter Luschny, Apr 20 2016

A256280 Expansion of phi(q^3)^4 / (phi(q) * phi(q^9)) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, 0, -2, 8, 0, 0, 4, 4, -4, 0, 0, -4, 0, 0, -2, 8, 4, 0, 8, 0, 0, 0, 0, -6, 8, 0, 0, 8, 0, 0, 4, 0, -4, 0, 4, -4, 0, 0, -4, 8, 0, 0, 0, 8, 0, 0, 0, -2, 12, 0, -4, 8, 0, 0, 0, 0, -4, 0, 0, -4, 0, 0, -2, 16, 0, 0, 8, 0, 0, 0, 4, -4, 8, 0, 0, 0, 0, 0, 8

Offset: 0

Michael Somos, Jun 02 2015

sign

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

			G.f. = 1 - 2*q + 4*q^2 - 2*q^4 + 8*q^5 + 4*q^8 + 4*q^9 - 4*q^10 - 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. A002175, A004018, A121444, A122856, A122865.

A := Basis( ModularForms( Gamma1(36), 1), 91); A[1] - 2*A[2] + 4*A[3] - 2*A[5] + 8*A[6] + 4*A[9] + 4*A[10] - 4*A[11] - 4*A[14] - 2*A[17] + 8*A[18] + 4*A[19];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^4 / (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9]), {q, 0, n}];

{a(n) = if( n<1, n==0, (-2)^(n%3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^10 * eta(x^9 + A) * eta(x^36 + A))^2 / (eta(x^2 + A)^5 * eta(x^3 + A)^8 * eta(x^12 + A)^8 * eta(x^18 + A)^5), n))};

Expansion of (eta(q) * eta(q^4) * eta(q^6)^10 * eta(q^9) * eta(q^36))^2 / (eta(q^2)^5 * eta(q^3)^8 * eta(q^12)^8 * eta(q^18)^5) in powers of q.
Euler transform of period 36 sequence [ -2, 3, 6, 1, -2, -9, -2, 1, 4, 3, -2, -3, -2, 3, 6, 1, -2, -6, -2, 1, 6, 3, -2, -3, -2, 3, 4, 1, -2, -9, -2, 1, 6, 3, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
a(3*n + 1) = -2 * A122865(n). a(3*n + 2) = 4 * A122856(n). a(4*n + 3) = 0.  a(4*n) = a(n). a(9*n) = A004018(n). a(9*n + 3) = a(9*n + 6) = 0. a(12*n + 1) = -2 * A002175(n). a(12*n + 5) = 8 * A121444(n).

A258034 Expansion of phi(q) * phi(q^9) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 03 2015

nonn

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

			G.f. = 1 + 2*q + 2*q^4 + 4*q^9 + 4*q^10 + 4*q^13 + 2*q^16 + 4*q^18 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A002175, A004018, A019670, A028601, A122856, A122865, A258322.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[5] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[19];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^9], {q, 0, n}];
a[ n_] := Which[ n < 1, Boole[n == 0], Mod[n, 3] == 2, 0, True, 2 DivisorSum[ n, If[ Mod[n/#, 9] > 0, 1, 2] KroneckerSymbol[ -4, #] &]]; (* Michael Somos, Jul 04 2015 *)

{a(n) = if( n<1, n==0, (n+1)%3 * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^9 + A) * eta(x^36 + A))^2, n))};

{a(n) = if( n<1, n==0, n%3==2, 0, 2 * sumdiv(n, d, if(n\d%9, 1, 2) * kronecker( -4, d)))}; /* Michael Somos, Jul 04 2015 */

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); (n%3 < 2) * 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))}; /* Michael Somos, Jul 04 2015 */

Expansion of eta(q^2)^5 * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^9) * eta(q^36))^2 in powers of q.
Euler transform of period 36 sequence [2, -3, 2, -1, 2, -3, 2, -1, 4, -3, 2, -1, 2, -3, 2, -1, 2, -6, 2, -1, 2, -3, 2, -1, 2, -3, 4, -1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A258322(n). a(4*n) = a(n).
a(3*n + 2) = a(4*n + 3) = a(8*n + 6) =  a(9*n + 3) = a(9*n + 6) = 0.
a(3*n + 1) = 2 * A122865(n).  a(6*n + 4) = 2 * A122856(n). a(9*n) = A004018(n). a(12*n + 1) = 2 * A002175(n).
a(2*n) = A028601(n). - Michael Somos, Jul 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 (A019670). - Amiram Eldar, Jan 29 2024

A258279 Expansion of psi(q)^2 * chi(-q^3)^2 in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, May 25 2015

sign

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

			G.f. = 1 + 2*q + q^2 - 2*q^4 - 2*q^5 + q^8 - 4*q^9 - 4*q^10 + 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. A002175, A004018, A089810, A121444, A258277, A258278.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, Pi/6, q]^2, {q, 0, n}];

{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 + A) * eta(x^6 + A)))^2, n))};

Expansion of eta(q^2)^4 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)^2) in powers of q.
Euler transform of period 6 sequence [ 2, -2, 0, -2, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 36 (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A002175.
G.f.: Product_{k>0} (1 - x^(2*k))^2 / (1 - x^k + x^(2*k))^2.
Convolution square of A089810.
a(2*n) = A258228(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = A258278(n). a(4*n + 3) = 0. a(6*n + 2) = A122865(n). a(6*n + 4) = -2 * A122856(n). a(12*n + 1) = 2 * A002175(n). a(12*n + 5) = -2 * A121444(n).
a(18*n) = A004018(n). a(18*n + 3) = a(18*n + 6) = a(18*n + 12) = 0.

A256276 Expansion of q * phi(q) * chi(q^3) * psi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jun 02 2015

nonn

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

			G.f. = q + 2*q^2 + q^4 + 4*q^5 + 2*q^8 + 2*q^10 + 2*q^13 + q^16 + 4*q^17 + ...

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. A122856, A122865, A256269.

A := Basis( ModularForms( Gamma1(36), 1), 89); A[2] + 2*A[3]
+ A[5] + 4*A[6] + 2*A[9] + 2*A[11] + 2*A[14] + A[17] + 4*A[18];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / (2^(1/2) q^(1/8)) QPochhammer[ -q^3, q^6] EllipticTheta[ 3, 0, q], {q, 0, n}];

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^2 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^3 + A) * eta(x^12 + A) * eta(x^18 + A)), n))};

Expansion of eta(q^2)^5 * eta(q^6)^2 * eta(q^9) * eta(q^36) / (eta(q)^2 * eta(q^4)^2 * eta(q^3) * eta(q^12) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 2, -3, 3, -1, 2, -4, 2, -1, 2, -3, 2, -1, 2, -3, 3, -1, 2, -4, 2, -1, 3, -3, 2, -1, 2, -3, 2, -1, 2, -4, 2, -1, 3, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A256269.
a(3*n) = a(4*n + 3) = 0. a(3*n + 1) = A122865(n). a(3*n + 2) = 2 * A122856(n). a(4*n + 1) = a(n). a(4*n) = a(n). a(6*n + 2) = 2 * A122865(n). a(6*n + 4) = A122856(n).

A259761 Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 04 2015

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

			G.f. = 1 + 2*x + 2*x^2 + 2*x^4 + 4*x^5 + 2*x^8 + 4*x^9 + 4*x^10 + 4*x^13 + ...

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. A004018, A019685, A122856.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(36), 1), 87); A[1] + 2*A[2] + 2*A[3] + 2*A[5] + 4*A[6] + 2*A[9] + 4*A[10] + 4*A[11] + 4*A[14] + 2*A[17] + 4*A[18] + 4*A[19];

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^2 + EllipticTheta[ 3, 0, x^9]^2) / 2, {x, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], 2 Times @@ ( Which[ # < 3, 1, # == 3, 1 + (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, (1 + (-1)^#2) / 2 ] & @@@ FactorInteger[n])];

{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==2, 1, p==3, 1 + (-1)^e, p%12>6, (1 + (-1)^e) / 2, e+1)))};

phi(x) = 1 + 2*Sum_{m=1..oo} x^(m^2). - N. J. A. Sloane, Jan 30 2017
Expansion of phi(x) * phi(x^9) + 2 * x^2 * chi(x^3)^2 * psi(-x^9)^2 in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
a(n) = 2 * b(n) with a(0) = 1 and b() is multiplicative with b(2^e) = 1, b(3^e) = 1 + (-1)^e if e>0, b(p^e) = e+1 if p == 1, 5 (mod 12), (p^e) = (1 + (-1)^e)/2 if p == 7, 11 (mod 12).
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
a(2*n) = a(n). a(9*n) = A004018(n). a(6*n + 4) = 2 * A122856(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 5*Pi/9 = 1.745329... (= 100 * A019685). - Amiram Eldar, Dec 29 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

A281452 Expansion of f(x, x) * f(x^5, x^13) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 1, 4, 0, 2, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 1, 4, 2, 0, 2, 0, 0, 0, 2, 2, 2, 0, 0, 2, 0, 0, 3, 2, 0, 0, 2, 4, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 4, 0, 0, 0, 0, 5, 2, 0, 0, 2, 0, 0, 0, 4, 2, 0, 2, 2, 0, 0, 0, 2, 2

Offset: 0

Michael Somos, Jan 26 2017

nonn

			G.f. = 1 + 2*x + 2*x^4 + x^5 + 2*x^6 + 4*x^9 + x^13 + 4*x^14 + 2*x^16 + ...
G.f. = q^4 + 2*q^13 + 2*q^40 + q^49 + 2*q^58 + 4*q^85 + q^121 + 4*q^130 + ...

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, 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. A281451, A281453, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 4, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^18] QPochhammer[ -x^13, x^18] QPochhammer[ x^18], {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 4])];

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

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

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

f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 4*k)).
G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 - x^(18*k-13)) * (1 - x^(18*k-5)) * (1 - x^(18*k)).
a(n) = A122865(3*n + 1) = A122856(6*n + 2) = A258278(6*n + 2). a(n) = - A256269(9^n + 4). 4 * a(n) = A004018(9*n + 4).
a(n) = b(9*n + 4) with b = A002654, A035154, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 4) = with b = A105673, A105673, A122857, A258034, A259761. -2 * a(n) = b(9*n + 4) with b = A138949, A256280, A258292.
a(4*n) = A281453(n). a(8*n + 6) = 2 * A281490(n). a(16*n + 12) = A281451(n).
a(32*n + 4) = 2 * A281492(n). a(64*n + 28) = A281452(n). a(128*n + 60) = 2 * A281491(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/3 = 1.0471975... (A019670). - Amiram Eldar, Jan 20 2025

A281453 Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jan 26 2017

nonn

			G.f. = 1 + 2*x + 2*x^4 + x^7 + 2*x^8 + 2*x^9 + 3*x^11 + 2*x^12 + ...
G.f. = q + 2*q^10 + 2*q^37 + q^64 + 2*q^73 + 2*q^82 + 3*q^100 + ...

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. A000122, A002654, A004018, A035154, A091400, A105673, A113446, A122856, A122857.
Cf. A122864, A122865, A125061, A129448, A138949, A138950, A163746, A205808, A256276.
Cf. A256280, A258034, A258228, A258256, A258278, A258292, A259761.
Cf. A281451, A281452, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 1, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^7, x^18] QPochhammer[ -x^11, x^18] QPochhammer[ x^18], {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 1])];

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

{a(n) = if( n<0, 0, my(m = 9*n + 1, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 1 || k%9 == 8), s+=(j>0)+1)); s)};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 1); 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)))};

f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 2*k)).
G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-11)) * (1 + x^(18*k-7)) * (1 - x^(18*k)).
a(4*n + 2) = a(8*n + 5) = a(16*n + 3) = a(32*n + 31) = a(64*n + 55) = a(128*n + 39) = 0.
a(4*n + 3) = A281451(n). a(8*n + 1) = 2 * A281492(n). a(16*n + 7) = A281452(n). a(32*n + 15) = 2 * A281491(n). a(128*n + 103) = 2 * A281490(n).
a(n) = A122865(3*n) = A122856(6*n) = A258278(6*n) = a(64*n + 7). a(n) = -A256269(9*n + 1).
a(n) = b(9*n + 1) where b = A002654, A035154, A091400, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 1) where b = A105673, A122857, A258034, A259761. 2 * a(n) = - b(9*n+1) where b = A138949, A256280, A258292. 4 * a(n) = A004018(9*n + 1).
Convolution of A000122 and A205808.

A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 0, -2, -8, 0, 0, 4, -4, -4, 0, 0, 4, 0, 0, -2, -8, 4, 0, 8, 0, 0, 0, 0, 6, 8, 0, 0, -8, 0, 0, 4, 0, -4, 0, 4, 4, 0, 0, -4, -8, 0, 0, 0, -8, 0, 0, 0, 2, 12, 0, -4, -8, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, -2, -16, 0, 0, 8, 0, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 8

Offset: 0

Michael Somos, Jun 03 2015

sign

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

			G.f. = 1 + 2*q + 4*q^2 - 2*q^4 - 8*q^5 + 4*q^8 - 4*q^9 - 4*q^10 + 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. A002175, A104794, A121444, A122856, A122865, A256280, A258277, A258278.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^4 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^9]), {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^8 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^9 + A)^2), n))};

{a(n) = if( n<1, n==0, 2^(n%3) * (-1)^(n\3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

Expansion of eta(q^2) * eta(q^3)^8 * eta(q^18) / (eta(q)^2 * eta(q^6)^4 * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ 2, 1, -6, 1, 2, -3, 2, 1, -4, 1, 2, -3, 2, 1, -6, 1, 2, -2, ...].
a(n) = (-1)^n * A256280(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = 4 * A258278(n). a(4*n) = A256280(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = -2 * A122856(n). a(9*n) = A104794(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -8 * A121444(n).

cp's OEIS Frontend

A091400 a(n) = Product_{ odd primes p | n } (1 + Legendre(-1,p) ).

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A256280 Expansion of phi(q^3)^4 / (phi(q) * phi(q^9)) in powers of q where phi() is a Ramanujan theta function.

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A258034 Expansion of phi(q) * phi(q^9) in powers of q where phi() is a Ramanujan theta function.

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A258279 Expansion of psi(q)^2 * chi(-q^3)^2 in powers of q where psi(), chi() are Ramanujan theta functions.

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A256276 Expansion of q * phi(q) * chi(q^3) * psi(-q^9) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

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A259761 Expansion of (phi(x)^2 + phi(x^9)^2) / 2 in powers of x where phi() is a Ramanujan theta function.

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