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 1-4 of 4 results.

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).

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

Views

Author

Michael Somos, May 30 2013

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 + 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 + ...

Links

Crossrefs

Cf. A053692, A226192.

Programs

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

Formula

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).

A245432 Expansion of f(-q^3, -q^5)^2 / (psi(-q) * phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, -1, -2, 3, 4, -6, -8, 11, 15, -20, -26, 34, 44, -56, -72, 91, 114, -143, -178, 220, 272, -334, -408, 498, 605, -732, -884, 1064, 1276, -1528, -1824, 2171, 2580, -3058, -3616, 4269, 5028, -5910, -6936, 8124, 9498, -11088, -12922, 15034, 17468, -20264
Offset: 0

Views

Author

Michael Somos, Jul 21 2014

Keywords

Comments

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

Examples

			G.f. = 1 + q - q^2 - 2*q^3 + 3*q^4 + 4*q^5 - 6*q^6 - 8*q^7 + 11*q^8 + ...

Links

Crossrefs

Cf. A115671, A210063, A226192, A224216, A244526.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2, q^4] / QPochhammer[ q^4, q^8]^2)^2 QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8] / (QPochhammer[ q^1, q^8] QPochhammer[ q^7, q^8]), {q, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 2, 1, -4, 1, 2, -1][k%8 + 1]), n))};
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); (-1)^(n \ 2) * polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};

Formula

Expansion of (f(-q^3, -q^5) / f(-q^1, -q^7)) * (psi(q^4) / phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [ 1, -2, -1, 4, -1, -2, 1, 0, ...].
Convolution quotient of A244526 and A226192.
a(n) = (-1)^floor(n/2) * A115671(n).
a(n) = A224216(n) unless n=0. a(2*n+1) = A210063(n).

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).
Showing 1-4 of 4 results.

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