A143378 -id:A143378 - cp's OEIS frontend

A143377 Expansion of q^(-1/6) * eta(q)^2 * eta(q^4) / eta(q^2) in powers of q.

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

Offset: 0

Michael Somos, Aug 10 2008

sign

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

			G.f. = 1 - 2*x + x^4 + 2*x^5 - 3*x68 - 2*x^12 + 2*x^13 + 2*x^16 + 2*x^17 + ...
G.f. = q - 2*q^7 + q^25 + 2*q^31 - 3*q^49 - 2*q^73 + 2*q^79 + 2*q^97 + 2*q^103 + ...

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. A143378, A143379, A143380.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^4] EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Apr 07 2015 *)

{a(n) = my(A, p, e, x); if(n<0, 0, A = factor(6*n + 1); simplify( I^n * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p<5, 0, p%8==5 || p%24==23, !(e%2), p%8==3 || p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e ))))};

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

Expansion of phi(-x) * f(-x^4) = psi(-x) * f(-x) = psi(-x)^2 * chi(-x^2) = f(-x)^2 / chi(-x^2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -2, -1, -2, -2, ...].
a(n) = (-1)^(n / 2) * b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 8) or p == 23 (mod 24), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3 (mod 8) or p == 17 (mod 24) and p>3, b(p^e) = (e+1) * s^e if p == 1, 7 (mod 24) where p = x^2 + 6*y^2 and s = Kronecker(12, x) * (-1)^((p-1) / 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 4608^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143379.
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)).
a(n) = (-1)^n * A143380(n). a(4*n) = A143378(n). a(4*n + 1) = -2 * A143379(n). - Michael Somos, Apr 07 2015

A143380 Expansion of q^(-1/6) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 11 2008

sign

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

			G.f. = 1 + 2*x + x^4 - 2*x^5 - 3*x^8 - 2*x^12 - 2*x^13 + 2*x^16 - 2*x^17 + ...
G.f. = q + 2*q^7 + q^25 - 2*q^31 - 3*q^49 - 2*q^73 - 2*q^79 + 2*q^97 + ...

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. A143377, A143378, A143379.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^5 / (QPochhammer[ x]^2 QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)

{a(n)= my(A, p, e, x); if(n<0, 0, A = factor(6*n + 1); simplify( I^-n * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p<5, 0, p%8==5 || p%24==23, !(e%2), p%8==3 || p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e ))))};

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

Expansion of psi(x)^2 * chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 2, -3, 2, -2, ...].
a(n) = (-1)^(-n / 2) * b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 8) or p == 23 (mod 24), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3 (mod 8) or p == 17 (mod 24) and p>3, b(p^e) = (e+1) * s^e if p == 1, 7 (mod 24) where p = x^2 + 6*y^2 and s = Kronecker(12, x) * (-1)^((p-1) / 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 1152^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143378.
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: Product_{k>0} (1 - (-x)^k)^2 * (1 + x^(2*k)).
a(n) = (-1)^n * A143377(n). a(4*n) = A143378(n). a(4*n + 1) = 2 * A143379(n).

A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -5, -10, 9, 14, -10, 0, 14, 2, -11, -32, 0, 14, -9, 26, 2, 0, 16, -22, 14, 0, 0, 26, -17, -32, -22, -10, -34, 14, 45, 38, 0, -34, 38, -22, 2, 0, -10, 64, -20, 0, 0, 0, -23, -46, 16, 0, -46, -32, 26, -10, 25, 18, 0, 38, 50, 0, 0, -22, -80, 50, 0, 26, 2

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 + 2*x - 5*x^2 - 10*x^3 + 9*x^4 + 14*x^5 - 10*x^6 + 14*x^8 + ...
G.f. = q + 2*q^13 - 5*q^25 - 10*q^37 + 9*q^49 + 14*q^61 - 10*q^73 + ...

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. A000727, A106508, A143378, A187076, A187149.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] EllipticTheta[ 3, 0, x])^2, {x, 0, n}];

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

Expansion of q^(-1/12) * (eta(q^2)^5 / (eta(q) * eta(q^4)^2))^2 in powers of q.
Euler transform of period 4 sequence [ 2, -8, 2, -4, ...].
a(n) = A000727(2*n) = A187076(2*n) = A106508(4*n) = A187149(4*n).
Convolution square of A143378.

A128263 Coefficients of L-series for elliptic curve "17a4": y^2 + xy + y = x^3 - x^2 - x or y^2 + xy - y = x^3 - x^2.

Original entry on oeis.org

1, -1, 0, -1, -2, 0, 4, 3, -3, 2, 0, 0, -2, -4, 0, -1, 1, 3, -4, 2, 0, 0, 4, 0, -1, 2, 0, -4, 6, 0, 4, -5, 0, -1, -8, 3, -2, 4, 0, -6, -6, 0, 4, 0, 6, -4, 0, 0, 9, 1, 0, 2, 6, 0, 0, 12, 0, -6, -12, 0, -10, -4, -12, 7, 4, 0, 4, -1, 0, 8, -4, -9, -6, 2, 0, 4, 0, 0, 12, 2, 9, 6, -4, 0, -2, -4, 0, 0, 10, -6, -8, -4, 0, 0, 8, 0, 2, -9, 0, 1, -10, 0

Offset: 1

Michael Somos, Feb 21 2007

Unique cusp form of weight 2 for congruence group Gamma_0(17). - Michael Somos, Aug 11 2011

			G.f. = q - q^2 - q^4 - 2*q^5 + 4*q^7 + 3*q^8 - 3*q^9 + 2*q^10 - 2*q^13 - ...

Robin Visser, Table of n, a(n) for n = 1..10000
LMFDB, Elliptic curve with LMFDB label 17.a4 (Cremona label 17a4)

Basis( CuspForms( Gamma0(17), 2), 10) [1]; /* Michael Somos, May 27 2014 */

{a(n) = local(A, p, e, x, y, a0, a1); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==17, 1, a0=1; a1 = y = -if( p==2, 1, sum( x=0, p-1, kronecker( 4*x^3 - 3*x^2 - 2*x + 1, p))); for( i=2, e, x = y*a1 - p*a0; a0=a1; a1=x); a1))))};

{a(n) = if( n<1, 0, ellak( ellinit([ 1, -1, 1, -1, 0], 1), n))};

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 * eta(x^34 + A)^5 / (eta(x^2 + A) * eta(x^17 + A) * eta(x^68 + A)^2) - x^4 * eta(x^2 + A)^5 * eta(x^17 + A) * eta(x^68 + A)^2 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^34 + A)), n))}; /* Michael Somos, Jan 01 2009 */

CuspForms( Gamma0(17), 2, prec = 100).0; # Michael Somos, Aug 11 2011

a(n) is multiplicative with a(17^e) = 1, a(p^e) = a(p) * a(p^(e-1)) - p * a(p^(e-2)) where a(p) = p minus number of points of elliptic curve modulo p.
G.f. is a period 1 Fourier series which satisfies f(-1 / (17 t)) = 17 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(9*n) = -3 * a(n). a(9*n + 3) = a(9*n + 6) = 0.
Expansion of q * A(q) * B(q^17) - q^5 * A(q^17) * B(q) where A(), B() are the g.f. for A143379, A143378 respectively. - Michael Somos, Jan 01 2009
Expansion of eta(q) * eta(q^4)^2 * eta(q^34)^5 / (eta(q^2) * eta(q^17) * eta(q^68)^2) - eta(q^2)^5 * eta(q^17) * eta(q^68)^2 / (eta(q) * eta(q^4)^2 * eta(q^34)) in powers of q. - Michael Somos, Jan 01 2009

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A143377 Expansion of q^(-1/6) * eta(q)^2 * eta(q^4) / eta(q^2) in powers of q.

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A143380 Expansion of q^(-1/6) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)) in powers of q.

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A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

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A128263 Coefficients of L-series for elliptic curve "17a4": y^2 + x*y + y = x^3 - x^2 - x or y^2 + x*y - y = x^3 - x^2.

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A128263 Coefficients of L-series for elliptic curve "17a4": y^2 + xy + y = x^3 - x^2 - x or y^2 + xy - y = x^3 - x^2.