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.

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A278428 Series reversion of g.f. (1/2)*x*(-1; -x)_inf, where (a; q)_inf is the q-Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 2, 6, 17, 46, 128, 373, 1119, 3405, 10464, 32478, 101781, 321642, 1023512, 3276326, 10543100, 34088806, 110690682, 360810160, 1180195810, 3872588051, 12743937024, 42049240694, 139082885503, 461072582522, 1531697761470, 5098246648103, 17000237006441
Offset: 1

Views

Author

Vladimir Reshetnikov, Nov 21 2016

Keywords

Comments

(1/2)*x*(-1; -x)_inf is the g.f. for A081360 shifted right.

Links

Crossrefs

Cf. A081360, A109085, A171805, A181315, A255526.

Programs

  • Mathematica
    InverseSeries[x QPochhammer[-1, -x]/2 + O[x]^35][[3]]
(* Calculation of constant c: *) 1/Sqrt[(4/s^2 - s*Derivative[0, 2][QPochhammer][-1, -s]/r) * Pi] /. FindRoot[{2*r == s*QPochhammer[-1, -s], 2*r == s^2*Derivative[0, 1][QPochhammer][-1, -s]}, {r, 1/3}, {s, 1/2}, WorkingPrecision -> 120] (* Vaclav Kotesovec, Oct 03 2023 *)

Formula

a(n) ~ c * d^n / n^(3/2), where c = 0.1211369424750398272226454930396... and d = A318204 = 3.509754327949703340437273523375193698454789733931739911... - Vaclav Kotesovec, Nov 23 2016

A374064 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-1)).

Original entry on oeis.org

1, 0, -1, 0, 1, -1, -1, 1, 0, -1, 1, 0, -1, 1, 0, -2, 2, 1, -3, 1, 3, -3, 0, 3, -3, -1, 4, -3, -1, 5, -3, -3, 7, -3, -5, 7, -1, -7, 8, 0, -8, 8, 1, -11, 10, 3, -14, 9, 8, -17, 8, 10, -18, 6, 14, -22, 6, 19, -24, 1, 26, -26, -3, 30, -25, -9, 37, -27, -13, 42, -26, -23, 51, -25, -31, 56
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035386, A081360, A081362, A109389, A132463, A261612, A262928, A284315, A374065.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A262928(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132463(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A261612(n-k).

A374065 Expansion of Product_{k>=1} 1 / (1 + x^(3*k-2)).

Original entry on oeis.org

1, -1, 1, -1, 0, 0, 0, -1, 2, -2, 1, 0, -1, 0, 2, -3, 3, -1, -1, 1, 1, -4, 5, -3, 0, 2, 0, -4, 7, -6, 1, 3, -2, -3, 9, -10, 4, 3, -5, -1, 11, -15, 10, 1, -8, 3, 10, -20, 17, -3, -10, 9, 7, -24, 26, -10, -10, 15, 2, -27, 37, -21, -8, 22, -6, -28, 49, -36, -2, 30, -19, -24, 61, -56, 10, 35
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A035382, A081360, A081362, A109389, A132462, A261612, A262928, A284312, A374064.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(3 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 3] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A261612(k) * a(n-k).
a(n) = Sum_{k=0..n} A081362(k) * A132462(n-k).
a(n) = Sum_{k=0..n} A109389(k) * A262928(n-k).

A131729 Period 4: repeat [0, 1, -1, 1].

Original entry on oeis.org

0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1, -1, 1, 0, 1
Offset: 0

Views

Author

Paul Curtz, Sep 17 2007

Keywords

Examples

			G.f. = x - x^2 + x^3 + x^5 - x^6 + x^7 + x^9 - x^10 + x^11 + x^13 - x^14 + x^15 + ...

Links

Crossrefs

Cf. A081360, A209635 (Dirichlet inverse), A166486 (absolute values).

Programs

Formula

From Michael Somos, Apr 10 2011: (Start)
Expansion of x * (1 - x) * (1 - x^6) / ((1 - x^2) * (1 - x^3) * (1 - x^4)) = (x - x^2 + x^3) / (1 - x^4) in powers of x.
Euler transform of length 6 sequence [-1, 1, 1, 1, 0, -1].
Moebius transform is length 4 sequence [1, -2, 0, 1].
a(n) is multiplicative with a(2) = -1, a(2^e) = 0 if e>1, a(p^e)=1 if p>2.
E.g.f.: sinh(x) + (cos(x) - cosh(x)) / 2. a(n) = a(-n) = a(n+4) for all n in Z. a(2*n + 1) = 0. a(4*n + 2) = -1. a(4*n) = 0. (End)
G.f.: A081360 = Product_{k>0} (1 - x^k)^a(k). - Michael Somos, Feb 06 2012
G.f.: x*(1-x+x^2)/ ((1-x)*(x+1)*(x^2+1)). - R. J. Mathar, Nov 15 2007
a(n) = 1/4+(1/2)*cos(1/2*Pi*n)+3/4*(-1)^(1+n). - R. J. Mathar, Nov 15 2007
Dirichlet g.f. (1-2^(-s))^2*zeta(s). - R. J. Mathar, Apr 14 2011

A294408 Expansion of 1/(1 + Sum_{i>=1} q^(i^2)/Product_{j=1..i} (1 + q^(2*j))).

Original entry on oeis.org

1, -1, 1, 0, -2, 3, -2, -1, 6, -10, 8, 2, -19, 34, -30, -3, 60, -112, 106, -2, -188, 370, -373, 48, 586, -1226, 1307, -296, -1808, 4046, -4546, 1430, 5516, -13300, 15724, -6217, -16626, 43566, -54132, 25464, 49373, -142146, 185496, -100306, -143896, 461874, -632864, 384348, 409270, -1494356, 2150240
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 30 2017

Keywords

Comments

Convolution inverse of the 3rd order mock theta function phi(q) (A053250).

Links

Crossrefs

Cf. A053250, A081360, A294407.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/(1 + Sum[q^(i^2)/Product[1 + q^(2 j), {j, 1, i}], {i, 1, nmax}]), {q, 0, nmax}], q]

Formula

G.f.: 1/(1 + Sum_{i>=1} q^(i^2)/Product_{j=1..i} (1 + q^(2*j))).

A374076 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-1)).

Original entry on oeis.org

1, 0, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, -1, 1, -1, 0, 1, -1, 2, -1, -1, 1, -2, 2, 0, -1, 2, -3, 2, 0, -2, 3, -3, 2, 1, -3, 4, -4, 2, 2, -4, 5, -5, 1, 3, -6, 7, -5, 1, 5, -8, 8, -6, -1, 8, -10, 11, -6, -3, 10, -14, 12, -5, -6, 15, -17, 14, -4, -10, 19, -21, 15, -1, -15, 25, -25
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109700, A281243, A284317, A374064, A374065, A374077, A374078, A374079.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 4 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A374077 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-2)).

Original entry on oeis.org

1, 0, 0, -1, 0, 0, 1, 0, -1, -1, 0, 1, 1, -1, -1, -1, 2, 1, 0, -2, -1, 1, 2, 0, -2, -2, 2, 2, 1, -3, -2, 1, 4, 1, -3, -4, 2, 4, 3, -5, -5, 0, 7, 4, -4, -8, 0, 7, 8, -5, -9, -4, 10, 9, -3, -13, -5, 9, 14, -3, -14, -10, 12, 16, 1, -19, -12, 10, 23, 1, -20, -20, 13, 26, 8, -26
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109699, A281271, A284320, A374064, A374065, A374076, A374078, A374079.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 2)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 3 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A374078 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-3)).

Original entry on oeis.org

1, 0, -1, 0, 1, 0, -1, -1, 1, 1, -1, -1, 0, 1, 1, -1, -1, 0, 1, 1, -1, -2, 0, 2, 2, -2, -3, 1, 4, 1, -4, -3, 3, 4, 0, -5, -3, 4, 5, -1, -6, -3, 6, 6, -2, -8, -3, 8, 8, -5, -11, -2, 12, 8, -8, -13, 1, 15, 8, -12, -15, 3, 19, 7, -16, -17, 6, 23, 8, -22, -20, 11, 30, 5, -30, -22
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109698, A281272, A284319, A374064, A374065, A374076, A374077, A374079.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 3)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 2 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A374079 Expansion of Product_{k>=1} 1 / (1 + x^(5*k-4)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 0, 0, 0, 0, 0, -1, 2, -2, 2, -2, 1, 0, -1, 1, -1, 0, 2, -3, 4, -4, 3, -1, -1, 2, -3, 2, 1, -4, 6, -7, 7, -4, 0, 3, -5, 5, -2, -3, 8, -11, 12, -9, 3, 3, -8, 10, -7, 0, 8, -15, 19, -17, 9, 1, -10, 16, -15, 6, 7, -19, 28, -29, 20, -5, -11, 23, -26, 17, 1, -21
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 27 2024

Keywords

Crossrefs

Cf. A081360, A081362, A109697, A280454, A284314, A374064, A374065, A374076, A374077, A374078.

Programs

  • Mathematica
    nmax = 75; CoefficientList[Series[Product[1/(1 + x^(5 k - 4)), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[DivisorSum[k, (-1)^(k/#) # &, Mod[#, 5] == 1 &] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 75}]

A246579 G.f.: x^(k^2)/(mul(1-x^(2*i),i=1..k)*mul(1+x^(2*r-1),r=1..oo)) with k=3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1, 2, -3, 5, -7, 11, -15, 21, -29, 39, -52, 69, -90, 116, -150, 190, -241, 303, -379, 470, -583, 716, -878, 1071, -1302, 1575, -1902, 2285, -2739, 3273, -3899, 4631, -5489, 6486, -7647, 8996, -10557, 12363, -14450, 16853, -19618, 22798, -26441
Offset: 0

Views

Author

N. J. A. Sloane, Aug 31 2014

Keywords

References

  • Fulman, Jason. Random matrix theory over finite fields. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 51--85. MR1864086 (2002i:60012). See top of page 70, Eq. 2, with k=3.

Crossrefs

k=0,1,2 give (apart perhaps from signs) A081360, A038348, A096778. Cf. A246590.

Programs

  • Maple
    fU:=proc(k) local a,i,r;
a:=x^(k^2)/mul(1-x^(2*i),i=1..k);
a:=a/mul(1+x^(2*r-1),r=1..101);
series(a,x,101);
seriestolist(%);
end;
fU(3);
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