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 1-5 of 5 results.

A339336 Dirichlet g.f.: Product_{k>=2} (1 + k^(-s))^4.

Original entry on oeis.org

1, 4, 4, 10, 4, 20, 4, 24, 10, 20, 4, 60, 4, 20, 20, 51, 4, 60, 4, 60, 20, 20, 4, 156, 10, 20, 24, 60, 4, 116, 4, 100, 20, 20, 20, 206, 4, 20, 20, 156, 4, 116, 4, 60, 60, 20, 4, 360, 10, 60, 20, 60, 4, 156, 20, 156, 20, 20, 4, 396, 4, 20, 60, 190, 20, 116, 4, 60, 20, 116, 4, 580
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 30 2020

Keywords

Crossrefs

Cf. A022569, A045778, A328706, A339319, A339335, A339337, A339338, A339339, A339340, A339341.

Formula

a(p^k) = A022569(k) for prime p.

A022599 Expansion of Product_{m>=1} (1+q^m)^(-4).

Original entry on oeis.org

1, -4, 6, -8, 17, -28, 38, -56, 84, -124, 172, -232, 325, -448, 594, -784, 1049, -1388, 1796, -2320, 3005, -3864, 4912, -6216, 7877, -9940, 12430, -15488, 19309, -23972, 29580, -36408, 44766, -54876, 66978, -81536, 99150, -120272, 145374, -175344, 211242
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
McKay-Thompson series of class 12J for the Monster group.

Examples

			G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 17*x^4 - 28*x^5 + 38*x^6 - 56*x^7 + ...
T12J = 1/q - 4*q^5 + 6*q^11 - 8*q^17 + 17*q^23 - 28*q^29 + 38*q^35 + ...

References

  • T. J. I'a. Bromwich, Introduction to the Theory of Infinite Series, Macmillan, 2nd. ed. 1949, p. 116, q_2^4.

Links

Crossrefs

Cf. A007259, A022597, A022569.
Column k=4 of A286352.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
      -4*irem(d, 2)*d, d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, May 02 2014
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[x^2])^4, {x, 0, n}]; (* Michael Somos, Jul 05 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^2 + A))^4, n))};

Formula

Expansion of chi(-x)^4 in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/6) * (eta(q) / eta(q^2))^4 in powers of q.
Euler transform of period 2 sequence [ -4, 0, ...]. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(q) = (A(q^6) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (16 + u * v) - v^2. - Michael Somos, Apr 26 2008
Given G.f. A(x) then B(q) = A(q^6) / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 4 * v * (v + u^2) - w^2 * (v - u^2). - Michael Somos, Apr 26 2008
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 4 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A022569.
Convolution inverse is A022569. Convolution square of A022597. Convolution square is A007259.
a(n) ~ (-1)^n * exp(Pi*sqrt(2*n/3)) / (2 * 6^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(4/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 03 2017
G.f.: exp(-4*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A022600 Expansion of Product_{m>=1} (1+q^m)^(-5).

Original entry on oeis.org

1, -5, 10, -15, 30, -56, 85, -130, 205, -315, 465, -665, 960, -1380, 1925, -2651, 3660, -5020, 6775, -9070, 12126, -16115, 21220, -27765, 36235, -47101, 60810, -78115, 100105, -127825, 162391, -205530, 259475, -326565
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. Related to Expansion of Product_{m>=1} (1+q^m)^k: A022627 (k=-32), A022626 (k=-31), A022625 (k=-30), A022624 (k=-29), A022623 (k=-28), A022622 (k=-27), A022621 (k=-26), A022620 (k=-25), A007191 (k=-24), A022618 (k=-23), A022617 (k=-22), A022616 (k=-21), A022615 (k=-20), A022614 (k=-19), A022613 (k=-18), A022612 (k=-17), A022611 (k=-16), A022610 (k=-15), A022609 (k=-14), A022608 (k=-13), A007249 (k=-12), A022606 (k=-11), A022605 (k=-10), A022604 (k=-9), A007259 (k=-8), A022602 (k=-7), A022601 (k=-6), this sequence (k=-5), A022599 (k=-4), A022598 (k=-3), A022597 (k=-2), A081362 (k=-1), A000009 (k=1), A022567 (k=2), A022568 (k=3), A022569 (k=4), A022570 (k=5), A022571 (k=6), A022572 (k=7), A022573 (k=8), A022574 (k=9), A022575 (k=10), A022576 (k=11), A022577 (k=12), A022578 (k=13), A022579 (k=14), A022580 (k=15), A022581 (k=16), A022582 (k=17), A022583 (k=18), A022584 (k=19), A022585 (k=20), A022586 (k=21), A022587 (k=22), A022588 (k=23), A014103 (k=24), A022589 (k=25), A022590 (k=26), A022591 (k=27), A022592 (k=28), A022593 (k=29), A022594 (k=30), A022595 (k=31), A022596 (k=32), A025233 (k=48).
Column k=5 of A286352.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[1/(1 + x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
  • PARI
    x='x+O('x^50); Vec(prod(m=1, 50, (1 + x^m)^(-5))) \\ Indranil Ghosh, Apr 05 2017

Formula

a(n) ~ (-1)^n * 5^(1/4) * exp(Pi*sqrt(5*n/6)) / (2^(7/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 05 2017
G.f.: exp(-5*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 06 2018

A227033 Expansion of (phi(x) / f(-x^4))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 4, 0, 6, 16, 8, 0, 17, 40, 28, 0, 38, 96, 56, 0, 84, 204, 124, 0, 172, 400, 232, 0, 325, 760, 448, 0, 594, 1376, 784, 0, 1049, 2404, 1388, 0, 1796, 4096, 2320, 0, 3005, 6808, 3864, 0, 4912, 11072, 6216, 0, 7877, 17688, 9940, 0, 12430, 27792, 15488, 0
Offset: 0

Views

Author

Michael Somos, Jul 03 2013

Keywords

Comments

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

Examples

			G.f. = 1 + 4*x + 4*x^2 + 6*x^4 + 16*x^5 + 8*x^6 + 17*x^8 + 40*x^9 + 28*x^10 + ...
G.f. = 1/q + 4*q^2 + 4*q^5 + 6*q^11 + 16*q^14 + 8*q^17 + 17*q^23 + 40*q^26 + ...

Links

Crossrefs

Cf. A022569, A112160.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x] / QPochhammer[ x^4])^2, {x, 0, n}];
  • 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^4 + A)^3))^2, n))};

Formula

Expansion of q^(1/3) * (eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 4 sequence [4, -6, 4, 0, ...].
a(4*n + 3) = 0. a(2*n) = A112160(n). a(4*n + 1) = 4 * A022569(n).

A227175 Expansion of (phi(x) / f(-x^4))^4 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, 24, 32, 28, 80, 192, 192, 134, 408, 864, 800, 568, 1520, 3072, 2752, 1809, 4808, 9456, 8192, 5316, 13616, 26112, 22144, 13990, 35376, 66624, 55584, 34696, 86016, 159744, 131392, 80724, 198256, 363720, 295776, 180068, 436816, 793344, 638976, 384940
Offset: 0

Views

Author

Michael Somos, Jul 03 2013

Keywords

Comments

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

Examples

			1 + 8*x + 24*x^2 + 32*x^3 + 28*x^4 + 80*x^5 + 192*x^6 + 192*x^7 + 134*x^8 + ...
q^-2 + 8*q + 24*q^4 + 32*q^7 + 28*q^10 + 80*q^13 + 192*q^16 + 192*q^19 + ...

Links

Crossrefs

Cf. A022569, A227033.

Programs

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

Formula

Expansion of q^(2/3) * (eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3))^4 in powers of q.
Euler transform of period 4 sequence [ 8, -12, 8, 0, ...].
a(2*n + 1) = 8 * A022569(n). Convolution square of A227033.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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