cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 10 results.

A143379 Expansion of q^(-7/24) * eta(q) * eta(q^4)^2 / eta(q^2) in powers of q.

Original entry on oeis.org

1, -1, 0, -1, -1, 1, 1, 1, -1, 1, 0, 1, 0, 0, -2, -1, 0, 0, -1, 1, 1, -2, 0, 0, 0, 1, 1, 0, 2, 0, 1, -1, -1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, -1, -1, -1, 0, 0, 0, 1, 0, 1, 0, 1, -1, -1, 2, 0, -1, 1, -1, 1, 0, 3, 1, -1, 0, 0, 0, 1, -2, 0, 0, -1, -1, 0, -1, 0, 1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 0, -1, 0, -2, 0, 1, 2, 1, -1, 0, 2, 1, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Michael Somos, Aug 11 2008

Keywords

Comments

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

Examples

			G.f. = 1 - x - x^3 - x^4 + x^5 + x^6 + x^7 - x^8 + x^9 + x^11 - 2*x^14 - x^15 - x^18 + ...
G.f. = q^7 - q^31 - q^79 - q^103 + q^127 + q^151 + q^175 - q^199 + q^223 + ...

Links

Crossrefs

Cf. A000009, A134343, A143377, A143380.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4]^2 / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Jul 11 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Apr 07 2015 *)
  • PARI
    {a(n) = my(A, p, e, x); if( n<0, 0, n = n*4 + 1; A = factor(6*n + 1); simplify( I^n / -2 * 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))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A), n))};

Formula

Expansion of psi(x^2) * f(-x) = psi(-x) * f(-x^4) = chi(-x) * f(-x^4)^2 = psi(-x)^2 / chi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Apr 07 2015
Euler transform of period 4 sequence [ -1, 0, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 72^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143377.
G.f.: Product_{k>0} (1 - x^(4*k))^2 * (1 - x^(2*k-1)).
Convolution of A000009 and A134343. - Michael Somos, Jul 11 2012
-2 * a(n) = A143377(4*n + 1). 2 * a(n) = A143380(4*n + 1).
a(2*n) = A214302(n). a(2*n + 1) = - A214303(n). - Michael Somos, Jul 11 2012

A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -4, 6, -8, 13, -12, 14, -24, 18, -20, 32, -24, 31, -40, 30, -32, 48, -48, 38, -56, 42, -44, 78, -48, 57, -72, 54, -72, 80, -60, 62, -104, 84, -68, 96, -72, 74, -124, 96, -80, 121, -84, 108, -120, 90, -112, 128, -120, 98, -156, 102, -104, 192, -108, 110
Offset: 0

Views

Author

Michael Somos, Aug 10 2006

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 33 of the 74 eta-quotients listed in Table I of Martin (1996).

Examples

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 13*x^4 - 12*x^5 + 14*x^6 - 24*x^7 + ...
G.f. = q - 4*q^3 + 6*q^5 - 8*q^7 + 13*q^9 - 12*q^11 + 14*q^13 - 24*q^15 + ...

References

  • J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.179
  • Hardy, et al., Collected Papers of Srinivasa Ramanujan, p. 326, Question 359.

Links

Crossrefs

Cf. A001938, A008438, A097723, A112610, A134343, A258831, A258835.

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(16), 2), 110); A[2] - 4*A[4]; /* Michael Somos, Jun 10 2015 */
  • Mathematica
    a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^2 / (4 q^(1/2)), {q, 0, n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^4, {q, 0, n}]; (* Michael Somos, Oct 14 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[1, 2 n + 1]]; (* Michael Somos, Jun 15 2015 *)
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sigma(2*n + 1))};
  • Sage
    A = ModularForms( Gamma0(16), 2, prec=110).basis(); A[1] - 4*A[3]; # Michael Somos, Jun 27 2013

Formula

Expansion of q^(-1/2) * (eta(q) * eta(q^4) / eta(q^2))^4 in powers of q.
Expansion of q^(-1/2)/4 * k * k' * (K / (Pi/2))^2 in powers of q where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012
Euler transform of period 4 sequence [ -4, 0, -4, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^n, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 (mod 4), b(p^e) = (-1)^e * (p^(e+1) - 1) / (p - 1) if p == 3 (mod 4).
Given g.f. A(x), then B(x) = 4 * Integral_{0..x} A(x^2) dx = arcsin(4 * x * A001938(x^2)) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = sin(u + v) / 2 - sin((u - v) / 2). - Michael Somos, Oct 14 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 27 2013
G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)))^4.
G.f.: Sum_{k>0} -(-1)^k * (2*k - 1) * x^(k - 1) / (1 + x^(2*k - 1)).
G.f.: (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(4*k)))^4.
G.f.: (Sum_{k>0} (-1)^floor(k/2) * x^((k^2 - k)/2))^4.
G.f.: Sum_{k>0} (-1)^k * (2*k - 1) * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A008438(n). a(2*n) = A112610(n). a(2*n + 1) = -4 * A097723(n).
Convolution square of A134343. - Michael Somos, Jun 20 2012
a(3*n + 2) = 6 * A258831(n). a(4*n + 3) = -8 * A258835(n). - Michael Somos, Jun 11 2015

A204531 Expansion of phi(q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 0, 0, 0, -4, 0, 0, -4, 2, 0, 0, 0, -4, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, -4, 0, 0, 4, 0, 0, 0, 0, -4, 0, 0, -8, 4, 0, 0, 0, -4, 0, 0, 0, 2, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 4, 8, 0, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, -8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0
Offset: 0

Views

Author

Michael Somos, Jan 15 2012

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q - 4*q^5 - 4*q^8 + 2*q^9 - 4*q^13 + 4*q^16 + 4*q^17 + ...

Links

Crossrefs

Cf. A053692, A104794, A113407, A134343, A246950.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}]; (* Michael Somos, Jun 10 2015 *)
  • PARI
    {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^8 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==5) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, e+1, !(e%2))))};

Formula

Expansion of eta(q^2)^5 / (eta(q)^2 * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [ 2, -3, 2, -3, 2, -3, 2, -2, ...].
G.f.: Product_{k>0} (1 - x^(2*k))^5 / ((1 - x^k)^2 * (1 - x^(8*k))).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(8*n) = A104794(n). a(4*n + 1) = 2 * A134343(n).
a(n) = (-1)^n * A246950(n). a(8*n + 1) = 2 * A113407(n). a(8*n + 5) = -4 * A053692(n). - Michael Somos, Jun 10 2015

A246950 Expansion of phi(-q) * phi(-q^4) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 0, 4, 0, 0, -4, -2, 0, 0, 0, 4, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, -6, 0, 0, 0, 4, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, -8, -4, 0, 0, 0, 4, 0, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, -8, 0, 0, 0, 0, 0, 0, -4, -4, 0, 0, 0, 0, 0, 0, 8, -2, 0
Offset: 0

Views

Author

Michael Somos, Sep 08 2014

Keywords

Comments

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

Examples

			G.f. = 1 - 2*q + 4*q^5 - 4*q^8 - 2*q^9 + 4*q^13 + 4*q^16 - 4*q^17 + ...

Links

Crossrefs

Cf. A053692, A104794, A113407, A134343, A204531.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(64), 1), 85); A[1] - 2*A[2] + 4*A[6] - 4*A[9] - 2*A[10] + 4*A[14] + 4*A[17] - 4*A[18] - 6*A[26] + 4*A[30] - 4*A[35] + 4*A[36]; /* Michael Somos, Jun 21 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^4], {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 / (eta(x^2 + A) * eta(x^8 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * (-1)^(n%8==1) * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 2 * (e>2) * (-1)^(e<4), p%4==1, (e+1), !(e%2))))};

Formula

Expansion of f(-q, -q) * f(q, -q) in powers of q where f(,) is the Ramanujan general theta function. - Michael Somos, Jun 21 2015
Expansion of eta(q)^2 * eta(q^4)^2 / (eta(q^2) * eta(q^8)) in powers of q.
Euler transform of period 8 sequence [ -2, -1, -2, -3, -2, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A053692.
G.f.: Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k)).
a(n) = (-1)^n * A204531(n). a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0.
a(8*n) = A104794(n). a(4*n + 1) = - 2 * A134343(n).
a(8*n + 1) = -2 * A113407(n). a(8*n + 5) = 4 * A053692(n). - Michael Somos, Jun 10 2015

A209942 Expansion of (psi(-x) * phi(x)^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 14, 81, 238, 322, 0, -429, -82, 0, -2162, -3038, 1134, 2401, -2482, 0, 6958, 3332, 0, 1442, 0, 6561, 4508, -9758, 0, -1918, -18802, 0, -9362, -24638, 19278, 14641, -14756, 0, 0, 6562, 0, -1148, 33998, 26082, 20398, 0, 0, 28083, -49042, 0, 64078, -30268, 0
Offset: 0

Views

Author

Michael Somos, Mar 16 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 60 of the 74 eta-quotients listed in Table I of Martin (1996).

Examples

			G.f. = 1 + 14*x + 81*x^2 + 238*x^3 + 322*x^4 - 429*x^6 - 82*x^7 - 2162*x^9 + ...
G.f. = q + 14*q^5 + 81*q^9 + 238*q^13 + 322*q^17 - 429*q^25 - 82*q^29 + ...

Links

Crossrefs

Cf. A000143, A134343, A258771.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^19 / (QPochhammer[ x] QPochhammer[ x^4])^7)^2, {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^19 / (eta(x + A) * eta(x^4 + A) )^7 )^2, n))};

Formula

Expansion of q^(-1/4) * ( eta(q^2)^19 / (eta(q) * eta(q^4) )^7 )^2 in powers of q.
Euler transform of period 4 sequence [ 14, -24, 14, -10, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32768 (t/i)^5 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 * p^(2*e) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p^4 * b(p^(e-2)) otherwise.
a(9*n + 5) = a(9*n + 8) = 0. a(9*n + 2) = 81 * a(n). Convolution of A000143 and A134343.
Convolution square of A258771. - Michael Somos, Jun 09 2015

A259285 Expansion of psi(x^2) * f(x, x^7) in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 0, 1, 0, 0, 1, 1, 1, 2, 0, 1, 0, 0, 2, 1, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 1, 0, 2, 0, 1, 0, 1, 3, 0, 1, 0, 1, 3, 1, 0, 0, 0, 0, 1, 2, 1, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 3, 1, 0, 1, 0
Offset: 0

Views

Author

Michael Somos, Jun 23 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x + x^2 + x^3 + x^6 + 2*x^7 + x^9 + x^10 + 2*x^12 + 2*x^13 + ...
G.f. = q^13 + q^29 + q^45 + q^61 + q^109 + 2*q^125 + q^157 + q^173 + ...

Links

Crossrefs

Cf. A000161, A025426, A025435, A025441, A134343, A259287.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x^1, x^8] QPochhammer[ -x^2, x^8] QPochhammer[ -x^6, x^8] QPochhammer[ -x^7, x^8] QPochhammer[x^8]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 + x^(8 k - 1)) (1 + x^(8 k - 2)) (1 + x^(8 k - 6)) (1 + x^(8 k - 7)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];
  • PARI
    {a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 13); for(u = (s+3)\-8, (s-3)\8, if( issquare( m - (8*u + 3)^2, &x) && (x%8==2 || x%8==6), c++))); c};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, -1, 0, 0, 1, 0, -1, -1, 2, -1, -1, 0, 1, 0, 0, -1][k%16 + 1], 1 + x * O(x^n)), n))};

Formula

Number of solutions to 16*n + 13 = (8*u + 3)^2 + (8*v + 2)^2 where u,v in Z.
Euler transform of period 16 sequence [ 1, 0, 0, -1, 0, 1, 1, -2, 1, 1, 0, -1, 0, 0, 1, -2, ...].
a(9*n + 2) = A259287(n). a(9*n + 5) = a(9*n + 8) = 0.
-2 * a(n) = A134343(4*n + 3). a(n) = A000161(16*n + 13) = A025426(16*n + 13) = A025435(16*n + 13) = A025441(16*n + 13).

A259287 Expansion of psi(x^2) * f(x^3, x^5) in powers of x where psi(), f(, ) are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 3, 0, 0, 1, 1, 1, 1, 0, 0, 0, 2, 0, 0, 2, 1, 2, 0, 0, 1, 2, 0, 1, 1, 0, 1, 3, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 2, 2, 1, 1, 1, 0, 0, 1, 0, 1, 1, 2, 0, 2, 0, 2, 2, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Michael Somos, Jun 23 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^5 + x^6 + x^7 + x^9 + x^11 + x^12 + x^14 + ...
G.f. = q^5 + q^37 + q^53 + 2*q^85 + q^101 + q^117 + q^149 + q^181 + q^197 + ...

Links

Crossrefs

Cf. A000161, A025426, A025435, A025441, A134343, A259285.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^8] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ -x^6, x^8] QPochhammer[x^8]^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[(1 + x^(8 k - 2)) (1 + x^(8 k - 3)) (1 + x^(8 k - 5)) (1 + x^(8 k - 6)) (1 - x^(8 k))^2, {k, Ceiling[n/8]}], {x, 0, n}];
  • PARI
    {a(n) = my(m, s, x, c); if( n<0, 0, s = sqrtint(m = 16*n + 5); for(u = (s+1)\-8, (s-1)\8, if( issquare( m - (8*u + 1)^2, &x) && (x%8==2 || x%8==6), c++))); c};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^[ 2, 0, -1, -1, 1, -1, 0, 0, 2, 0, 0, -1, 1, -1, -1, 0][k%16 + 1], 1 + x * O(x^n)), n))};

Formula

Number of solutions to 16*n + 5 = (8*u + 1)^2 + (8*v + 2)^2 where u,v in Z.
Euler transform of period 16 sequence [ 0, 1, 1, -1, 1, 0, 0, -2, 0, 0, 1, -1, 1, 1, 0, -2, ...].
a(9*n + 1) = a(9*n + 4) = 0. a(9*n + 7) = A259285(n).
-2 * a(n) = A134343(4*n + 1). a(n) = A000161(16*n + 5) = A025426(16*n + 5) = A025435(16*n + 5) = A025441(16*n + 5).

A246862 Expansion of phi(x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Sep 05 2014

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + x^3 + 4*x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^9 + 2*x^12 + ...
G.f. = q + 2*q^17 + q^49 + 4*q^65 + q^81 + 2*q^97 + 2*q^113 + 4*q^145 + ...

Links

Crossrefs

Cf. A000122, A008441, A113407, A116604, A125079, A129447, A134343, A138741, A214264, A226192, A246863.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8], {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, issquare(16 * n + 1) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 1)))};

Formula

Euler transform of period 16 sequence [ 2, -3, 3, -1, 3, -4, 2, -2, 2, -4, 3, -1, 3, -3, 2, -2, ...].
Convolution of A000122 and A214264.
a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = A246863(n).
a(n) = A113407(2*n) = A226192(2*n) = A008441(4*n) = A134343(4*n) = A116604(8*n) = A125079(8*n) = A129447(8*n) = A138741(8*n).

A213791 Expansion of psi(-x)^6 in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 15, -26, 45, -66, 82, -120, 156, -170, 231, -276, 290, -390, 435, -438, 561, -630, 651, -780, 861, -842, 1020, -1170, 1095, -1326, 1431, -1370, 1716, -1740, 1682, -2016, 2145, -2132, 2415, -2550, 2353, -2850, 3120, -2810, 3321, -3486, 3285, -3906, 4005
Offset: 0

Views

Author

Michael Somos, Jun 20 2012

Keywords

Comments

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

Examples

			G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 45*x^4 - 66*x^5 + 82*x^6 - 120*x^7 + ...
G.f. = q^3 - 6*q^7 + 15*q^11 - 26*q^15 + 45*q^19 - 66*q^23 + 82*q^27 + ...

References

  • J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. X
  • J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.176

Links

Crossrefs

Cf. A008440, A134343.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^2, x^4])^6, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) / eta(x^2 + A) )^6, n))};

Formula

Expansion of q^(-3/4) * ( eta(q) * eta(q^4) / eta(q^2) )^6 in powers of q.
Expansion of -1/(8 * r) * ( 1^2 * r^1 / (1 + q) - 3^2 * q^(3/4) / (1 + q^3) - 5^2 * r^5 / (1 + q^5) + 7^2 * q^(7/4) / (1 + q^7) + 9^2 * r^9 / (1 + q^9) - ...) in powers of q where r = q^(3/4) [Glaisher 1876].
Expansion of q^(-1/4) * ( sqrt(k * k') * K / Pi )^3 in powers of q where k, k', K are Jacobi elliptic functions. [Jacobi 1828, p. 108 quoted in Glaisher 1876, p. 176].
Euler transform of period 4 sequence [ -6, 0, -6, -6, ...].
G.f.: (Sum_{k>0} (-x)^((k^2 - k)/2))^6.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A008440(n). Convolution cube of A134343.

A246863 Expansion of phi(x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 2, 0, 2, 2, 0, 1, 2, 2, 3, 4, 0, 0, 2, 0, 4, 2, 0, 2, 0, 0, 1, 4, 0, 2, 6, 1, 2, 0, 0, 4, 2, 0, 0, 2, 4, 2, 2, 0, 0, 0, 0, 4, 0, 1, 4, 2, 0, 4, 2, 0, 3, 2, 2, 0, 4, 0, 2, 2, 0, 4, 0, 2, 2, 2, 0, 0, 2, 0, 2, 4, 0, 0, 2, 0, 3, 4, 0, 0, 2, 4, 2, 0, 0, 3, 4
Offset: 0

Views

Author

Michael Somos, Sep 05 2014

Keywords

Comments

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

Examples

			G.f. = 1 + 3*x + 2*x^2 + 2*x^4 + 2*x^5 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
G.f. = q^9 + 3*q^25 + 2*q^41 + 2*q^73 + 2*q^89 + q^121 + 2*q^137 + 2*q^153 + ...

Links

Crossrefs

Cf. A000122, A008441, A113407, A116604, A125079, A129447, A134343, A138741,A214263, A226192, A246862.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^1, x^8] QPochhammer[ -x^7, x^8] QPochhammer[ x^8], {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, issquare(16 * n + 9) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 9)))};

Formula

Euler transform of period 16 sequence [ 3, -4, 2, -1, 2, -3, 3, -2, 3, -3, 2, -1, 2, -4, 3, -2, ...].
Convolution of A000122 and A214263.
a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = A246862(n).
a(n) = A113407(2*n + 1) = - A226192(2*n + 1) = A008441(4*n + 2) = A134343(4*n + 2) = A116604(8*n + 4) = A125079(8*n + 4) = A129447(8*n + 4) = A138741(8*n + 4).
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