A258278 -id:A258278 - cp's OEIS frontend

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

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).

A256282 Expansion of f(-q^3) * psi(q^3)^3 / (psi(q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

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 - q + q^2 + q^4 - 2*q^5 + q^8 - 4*q^9 + 2*q^10 - 2*q^13 + q^16 + ...

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. A258256, A258277, A258278.

a[ n_] := SeriesCoefficient[ q^(1/8) QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3 / (2 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q^(9/2)]), {q, 0, n}];
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, {1, 2, -1, 0}[[Mod[#, 4, 1]]] If[ Divisible[#, 9], 4, 1] (-1)^(Boole[Mod[#, 8] == 6] + #) &]]; (* Michael Somos, Jun 06 2015 *)
a[ n_] := If[ n < 1, Boole[ n==0 ], -Times @@ (Which[ # == 1, 1, # == 2, -1, Mod[#, 4] == 1, #2 + 1, True, If[# == 3, 4, 1] Mod[#2 + 1, 2]] & @@@ FactorInteger[n])]; (* Michael Somos, Jun 06 2015 *)

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

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

Expansion of eta(q) * eta(q^6)^6 * eta(q^9) / (eta(q^2)^2 * eta(q^3)^2 * eta(q^18)^2) in powers of q.
Euler transform of period 18 sequence [ -1, 1, 1, 1, -1, -3, -1, 1, 0, 1, -1, -3, -1, 1, 1, 1, -1, -2, ...].
Moebius transform is a period 72 sequence.
a(n) = (-1)^n * A258256(n). a(2*n) = A258256(n). a(3*n + 1) = - A258277(n). a(3*n + 2) = A258278(n). a(4*n + 3) = 0.
a(n) = -b(n) where b() is multiplicative with a(0) = 1,  b(2^e) = -1 if e>0, b(3^e) = 2 * (1 + (-1)^e), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4). - Michael Somos, Jun 06 2015

A257900 Expansion of 1/2 - (phi(-q)^2 + phi(-q^9)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, May 25 2015

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

			G.f. = q - q^2 - q^4 + 2*q^5 - q^8 + 2*q^9 - 2*q^10 + 2*q^13 - q^16 + ...

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. A113652, A258277, A258278, A258322.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ 1/2 - (EllipticTheta[ 4, 0, q]^2 + EllipticTheta[ 4, 0, q^9]^2) / 4, {q, 0, n}]; (* Michael Somos, Jun 02 2015 *)

{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%4==3, if( p>3, 1, 2) * (1-e%2), e+1)))};

Expansion of 1/2 - (eta(q)^4 * eta(q^18)^2 + eta(q^2)^2 * eta(q^9)^4) / (2 * eta(q^2) * eta(q^18))^2 in powers of q. - Michael Somos, Jun 02 2015
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 1 + (-1)^e if e>0, a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4), a(p^e) = e+1 if p == 1 (mod 4).
a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(9*n) = 2 * A113652(n). a(9*n + 3) = a(9*n + 6) = 0.
-2 * a(n) = A258322(n) unless n = 0 or n == 2 (mod 3).
Sum_{k=1..n} abs(a(k)) ~  (5*Pi/18) * n. - Amiram Eldar, Jan 29 2024

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A256282 Expansion of f(-q^3) * psi(q^3)^3 / (psi(q) * psi(q^9)) in powers of q where psi(), f() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A257900 Expansion of 1/2 - (phi(-q)^2 + phi(-q^9)^2) / 4 in powers of q where phi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula