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.

A299108 Expansion of 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).

Original entry on oeis.org

1, 1, 3, 9, 27, 79, 231, 675, 1971, 5755, 16805, 49071, 143289, 418411, 1221781, 3567663, 10417761, 30420401, 88829145, 259385701, 757419669, 2211704625, 6458291945, 18858546645, 55067931981, 160801210705, 469547855419, 1371104033121, 4003694720243
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 02 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A288515.
Cf. A015128, A032803, A067687, A299105, A299106.

Programs

  • Maple
    S:= series(1/(1-x/JacobiTheta4(0,x)),x,51):
seq(coeff(S,x,n),n=0..50); # Robert Israel, Feb 02 2018
  • Mathematica
    nmax = 28; CoefficientList[Series[1/(1 - x Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x/EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
nmax = 28; CoefficientList[Series[1/(1 - x QPochhammer[-x, x]/QPochhammer[x, x]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Product_{k>=1} (1 + x^k)/(1 - x^k)).
G.f.: 1/(1 - x/theta_4(x)), where theta_4() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A015128(k-1)*a(n-k).
a(n) ~ c * d^n, where d = 2.9200517419026569743994130834319365190407162724411912701937027582419975778... is the root of the equation EllipticTheta(4, 0, 1/d) * d = 1 and c = 0.372842695601022868809531452599286285949969156503576039087883242107... = 2*Log[r]*QPochhammer[r] / (2*QPochhammer[r] * (Log[1 - r] + Log[r] + QPolyGamma[1, r]) + r*Log[r] * (r * Derivative[0, 1][QPochhammer][-1, r] - 2*Derivative[0, 1][QPochhammer][r, r])), where r = 1/d. Equivalently, c = EllipticTheta[4, 0, r]^2 / (r *(EllipticTheta[4, 0, r] - r * Derivative[0, 0, 1][EllipticTheta][4, 0, r])). - Vaclav Kotesovec, Feb 03 2018, updated Mar 31 2018

A302018 Expansion of 1/(1 - x*(1 + theta_3(x))/2), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 1, 2, 3, 5, 9, 15, 26, 44, 75, 129, 220, 377, 644, 1101, 1883, 3219, 5506, 9414, 16098, 27527, 47069, 80488, 137630, 235343, 402427, 688134, 1176685, 2012085, 3440591, 5883279, 10060183, 17202533, 29415676, 50299693, 86010564, 147074801, 251492331, 430042340, 735356089, 1257431006
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 30 2018

Keywords

Links

Crossrefs

Antidiagonal sums of A045847.
Cf. A010052, A032803, A181649, A302019.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[1/(1 - x (1 + EllipticTheta[3, 0, x])/2), {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[1/(1 - x Sum[x^k^2, {k, 0, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: 1/(1 - x*Sum_{k>=0} x^(k^2)).
a(0) = 1; a(n) = Sum_{k=1..n} A010052(k-1)*a(n-k).

A307901 Expansion of 1/(1 - x * theta_4(x)), where theta_4() is the Jacobi theta function.

Original entry on oeis.org

1, 1, -1, -3, -1, 7, 11, -5, -33, -25, 53, 123, 9, -297, -363, 323, 1273, 657, -2415, -4407, 957, 12069, 11465, -16887, -47915, -12939, 104431, 152029, -85529, -476579, -333905, 803237, 1752799, 11597, -4349949, -5019855, 5068735, 18311655, 8392559, -35953969
Offset: 0

Views

Author

Ilya Gutkovskiy, May 04 2019

Keywords

Links

Crossrefs

Cf. A002448, A015128, A032803, A299108.

Programs

  • Maple
    S:= series(1/(1-x*JacobiTheta4(0,x)),x,101):
seq(coeff(S,x,j),j=0..100);  # Robert Israel, Nov 03 2019
  • Mathematica
    nmax = 39; CoefficientList[Series[1/(1 - x EllipticTheta[4, 0, x]), {x, 0, nmax}], x]
nmax = 39; CoefficientList[Series[1/(1 - x Product[(1 - x^k)/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]

Formula

G.f.: Sum_{k>=0} x^k * theta_4(x)^k.
G.f.: 1/(1 - x * Sum_{k=-oo..oo} (-1)^k * x^(k^2)).
G.f.: 1/(1 - x * Product_{k>=1} (1 - x^k)/(1 + x^k)).

A329971 Expansion of 1 / (1 - 2 * Sum_{k>=1} x^(k^2)).

Original entry on oeis.org

1, 2, 4, 8, 18, 40, 88, 192, 420, 922, 2024, 4440, 9736, 21352, 46832, 102720, 225298, 494144, 1083804, 2377112, 5213736, 11435312, 25081112, 55010496, 120654744, 264632554, 580419672, 1273036832, 2792156864, 6124049048, 13431901808, 29460245120, 64615275940
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 26 2019

Keywords

Crossrefs

Cf. A000122, A004402, A006456, A025192, A032803, A240944, A279225, A279226, A280542, A320654.

Programs

  • Mathematica
    nmax = 32; CoefficientList[Series[1/(1 - 2 Sum[x^(k^2), {k, 1, Floor[Sqrt[nmax]] + 1}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - EllipticTheta[3, 0, x]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[SquaresR[1, k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]

Formula

G.f.: 1 / (2 - theta_3(x)), where theta_3() is the Jacobi theta function.
a(0) = 1; a(n) = Sum_{k=1..n} A000122(k) * a(n-k).
Showing 1-4 of 4 results.

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