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.

A373221 Expansion of Product_{i>=1, j>=0} (1 + x^(i * 7^j)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 14, 18, 22, 28, 34, 41, 50, 60, 72, 86, 105, 124, 146, 174, 204, 240, 282, 332, 386, 450, 524, 606, 703, 812, 940, 1082, 1243, 1428, 1636, 1873, 2140, 2448, 2788, 3172, 3610, 4096, 4646, 5264, 5963, 6737, 7607, 8584, 9668, 10887, 12244, 13765, 15451, 17328
Offset: 0

Views

Author

Seiichi Manyama, May 28 2024

Keywords

Comments

This sequence is different from A093950. The first difference occurs at a(50) = 6737, whereas A093950(50) = 6736.

Crossrefs

Cf. A000041, A174065, A327726, A373219, A373220.
Cf. A000009, A093950, A373217.

Programs

  • PARI
    my(N=60, x='x+O('x^N)); Vec(prod(k=1, N, (1+x^k)^(valuation(k, 7)+1)))

Formula

G.f.: Product_{k>=1} (1 + x^k)^A373217(k).
Let A(x) be the g.f. of this sequence, and B(x) be the g.f. of A000009, then B(x) = A(x)/A(x^7).

A102314 McKay-Thompson series of class 42C for the Monster group.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -2, 3, -2, 3, -3, 4, -4, 4, -6, 7, -7, 7, -9, 10, -12, 13, -14, 17, -18, 19, -22, 26, -28, 29, -34, 38, -41, 44, -50, 57, -60, 65, -72, 81, -86, 94, -105, 114, -124, 133, -146, 161, -174, 187, -204, 224, -240, 258, -282, 309, -332, 354, -386, 419, -450, 481, -524, 569, -606, 651, -703
Offset: 0

Views

Author

Michael Somos, Jan 03 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(x), the second term of the left side of Cayley's identity is -A(q). - Michael Somos, Dec 03 2013

Examples

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - 2*x^7 + 3*x^8 - 2*x^9 + 3*x^10 - 3*x^11 + ...
T42C = 1/q - q^2 - q^8 + q^11 - q^14 + q^17 - 2*q^20 + 3*q^23 - 2*q^26 + ...

References

  • A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.

Links

Crossrefs

Cf. A093950, A112212, A193826, A193883.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, n}]; (* Michael Somos, Aug 06 2011 *)
a[ n_] := SeriesCoefficient[ 1 / ( Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}] ), {x, 0, n}]; (* Michael Somos, Aug 06 2011 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)), n))};

Formula

Expansion of chi(-x) * chi(-x^7) in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/3) * eta(q) * eta(q^7) / (eta(q^2) * eta(q^14)) in powers of q.
Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, ...].
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^2*v - 2*u.
G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A093950.
G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^(7*k))).
a(n) = (-1)^n * A112212(n). a(2*n + 1) = - A093950(n). a(4*n) = A193826(n). a(4*n + 2) = A193883(n).
Convolution inverse is A093950.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

A112212 McKay-Thompson series of class 84C for the Monster group.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 3, 2, 3, 3, 4, 4, 4, 6, 7, 7, 7, 9, 10, 12, 13, 14, 17, 18, 19, 22, 26, 28, 29, 34, 38, 41, 44, 50, 57, 60, 65, 72, 81, 86, 94, 105, 114, 124, 133, 146, 161, 174, 187, 204, 224, 240, 258, 282, 309, 332, 354, 386, 419, 450, 481, 524, 569, 606
Offset: 0

Views

Author

Michael Somos, Aug 28 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(x), the first term of the left side of Cayley's identity is A(q). - Michael Somos, Dec 03 2013

Examples

			G.f. = 1 + x + x^3 + x^4 + x^5 + x^6 + 2*x^7 + 3*x^8 + 2*x^9 + 3*x^10 + ...
T84C = 1/q + q^2 + q^8 + q^11 + q^14 + q^17 + 2*q^20 + 3*q^23 + 2*q^26 + ...

References

  • A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.

Links

Crossrefs

Cf. A093950, A102314.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^7, x^14], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}] Product[ 1 + x^k, {k, 7, n, 14}], {x, 0, n}]; (* Michael Somos, Dec 03 2013 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^14 + A)^2 / (eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A)), n))}; /* Michael Somos, Dec 03 2013 */

Formula

Expansion of q^(1/3) * eta(q^2)^2 * eta(q^14)^2 / (eta(q) * eta(q^4) * eta(q^7) * eta(q^28)) in powers of q. - Michael Somos, Dec 03 2013
Euler transform of period 28 sequence [1, -1, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, 0, 2, -1, 1, 0, 1, -1, 1, 0, ...]. - Michael Somos, Dec 03 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (28 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 03 2013
G.f.: Product_{k>0} (1 + x^(2*k - 1)) * (1 + x^(14*k - 7)). - Michael Somos, Dec 03 2013
a(n) = (-1)^n * A102314(n). a(2*n + 1) = A093950(n). - Michael Somos, Dec 03 2013
a(n) ~ exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

A246762 Expansion of 1 / (chi(x) * chi(x^7)) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -2, 2, -3, 4, -6, 7, -9, 12, -14, 18, -22, 28, -34, 41, -50, 60, -72, 86, -105, 124, -146, 174, -204, 240, -282, 332, -386, 450, -524, 606, -703, 812, -940, 1082, -1243, 1428, -1636, 1873, -2140, 2448, -2788, 3172, -3610, 4096, -4646, 5264, -5962
Offset: 0

Views

Author

Michael Somos, Sep 02 2014

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 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 6*x^7 + 7*x^8 - 9*x^9 + ...
G.f. = q - q^4 + q^7 - 2*q^10 + 2*q^13 - 3*q^16 + 4*q^19 - 6*q^22 + 7*q^25 + ...

Links

Crossrefs

Cf. A093950, A112212.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, n}] Product[ 1 + (-x)^k, {k, 7, n, 7}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] QPochhammer[ x^7, -x^7], {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/3)* eta[q]*eta[q^4]*eta[q^7]*eta[q^28]/(eta[q^2]*eta[q^14])^2, {q,0,60}],q]; Table[a[[n]], {n,1,50}] (* G. C. Greubel, Jul 04 2018 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)) * prod( k=1, n\7, 1 + (-x)^(7*k), 1 + x * O(x^n)), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A) / (eta(x^2 + A) * eta(x^14 + A))^2, n))};

Formula

Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^7) * eta(q^28) / (eta(q^2) * eta(q^14))^2 in powers of q.
Euler transform of period 28 sequence [ -1, 1, -1, 0, -1, 1, -2, 0, -1, 1, -1, 0, -1, 2, -1, 0, -1, 1, -1, 0, -2, 1, -1, 0, -1, 1, -1, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 2 * (u*v)^2 * (1 - u*v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 + (-x)^k) * (1 + (-x)^(7*k)).
a(n) = (-1)^n * A093950(n).
Convolution inverse of A112212.
Showing 1-4 of 4 results.

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