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 1-8 of 8 results.

A300415 Expansion of Product_{k>=2} (1 + x^k)/(1 - x^k).

Original entry on oeis.org

1, 0, 2, 2, 4, 6, 10, 14, 22, 32, 46, 66, 94, 130, 182, 250, 340, 462, 622, 830, 1106, 1462, 1922, 2518, 3282, 4256, 5502, 7082, 9078, 11602, 14774, 18746, 23722, 29922, 37630, 47202, 59044, 73662, 91682, 113830, 140994, 174262, 214906, 264462, 324802, 398110, 487018, 594694
Offset: 0

Views

Author

Ilya Gutkovskiy, Mar 05 2018

Keywords

Comments

Convolution of the sequences A002865 and A025147.
Also number of overpartitions of n without a 1. - George Beck, Jan 25 2021

Links

Crossrefs

Cf. A002865, A015128, A025147, A207641, A211971.

Programs

  • Maple
    g:= (1-x)/((1+x)*JacobiTheta4(0,x)):
S:=series(g,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Mar 05 2018
  • Mathematica
    nmax = 47; CoefficientList[Series[Product[(1 + x^k)/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x]
nmax = 47; CoefficientList[Series[(1 - x)/((1 + x) EllipticTheta[4, 0, x]), {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=2} (1 + x^k)/(1 - x^k).
G.f.: (1 - x)/((1 + x)*theta_4(x)), where theta_4() is the Jacobi theta function.
a(n) ~ Pi * exp(Pi*sqrt(n)) / (32*n^(3/2)). - Vaclav Kotesovec, Mar 05 2018

A385088 G.f.: Sum_{k>=0} x^k * Product_{j=1..2*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 13, 23, 39, 63, 101, 159, 243, 367, 547, 801, 1161, 1665, 2359, 3315, 4621, 6385, 8761, 11941, 16165, 21757, 29121, 38761, 51337, 67673, 88793, 116009, 150949, 195629, 252595, 324987, 416675, 532483, 678333, 861489, 1090913, 1377553, 1734761, 2178883
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

Links

Crossrefs

Cf. A035294, A192433.
Cf. A207641, A385089, A385090, A385091, A385092.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 2*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(2*k))*(1 - x^(2*k - 1))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(2*k))*(1 + x^(2*k - 1))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(n)) / (16 * n^(3/4)).

A385089 G.f.: Sum_{k>=0} x^k * Product_{j=1..3*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 27, 47, 79, 127, 199, 307, 465, 695, 1025, 1493, 2151, 3069, 4337, 6075, 8441, 11639, 15933, 21667, 29281, 39337, 52555, 69849, 92375, 121595, 159347, 207939, 270259, 349911, 451377, 580223, 743341, 949241, 1208415, 1533763, 1941111, 2449841, 3083637
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

Links

Crossrefs

Cf. A035295, A385067.
Cf. A207641, A385088, A385090, A385091, A385092.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(3*k))*(1 - x^(3*k - 1))*(1 - x^(3*k - 2))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(3*k))*(1 + x^(3*k - 1))*(1 + x^(3*k - 2))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/3) * exp(Pi*sqrt(n)) / (3 * 2^(8/3) * Pi^(2/3) * n^(2/3)).

A385090 G.f.: Sum_{k>=0} x^k * Product_{j=1..4*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 51, 87, 143, 227, 353, 537, 803, 1185, 1727, 2489, 3551, 5021, 7039, 9791, 13521, 18541, 25261, 34207, 46051, 61655, 82113, 108815, 143517, 188433, 246343, 320725, 415931, 537377, 691791, 887517, 1134863, 1446549, 1838235, 2329147, 2942849, 3708165
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

Links

Crossrefs

Cf. A035296, A385068.
Cf. A207641, A385088, A385089, A385091, A385092.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 4*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(4*k))*(1 - x^(4*k - 1))*(1 - x^(4*k - 2))*(1 - x^(4*k - 3))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(4*k))*(1 + x^(4*k - 1))*(1 + x^(4*k - 2))*(1 + x^(4*k - 3))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/4) * exp(Pi*sqrt(n)) / (2^(9/2) * Pi^(3/4) * n^(5/8)).

A385091 G.f.: Sum_{k>=0} x^k * Product_{j=1..5*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 53, 91, 151, 243, 381, 585, 881, 1305, 1907, 2753, 3931, 5559, 7793, 10835, 14955, 20501, 27921, 37801, 50889, 68139, 90777, 120353, 158827, 208683, 273037, 355791, 461839, 597273, 769661, 988411, 1265149, 1614215, 2053297, 2604113, 3293281, 4153407
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

Links

Crossrefs

Cf. A035297, A385069.
Cf. A207641, A385088, A385089, A385090, A385092.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 5*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(5*k))*(1 - x^(5*k - 1))*(1 - x^(5*k - 2))*(1 - x^(5*k - 3))*(1 - x^(5*k - 4))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(5*k))*(1 + x^(5*k - 1))*(1 + x^(5*k - 2))*(1 + x^(5*k - 3))*(1 + x^(5*k - 4))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/5) * exp(Pi*sqrt(n)) / (5 * 2^(12/5) * Pi^(4/5) * n^(3/5)).

A385092 G.f.: Sum_{k>=0} x^k * Product_{j=1..6*k} (1 + x^j)/(1 - x^j).

Original entry on oeis.org

1, 1, 3, 7, 15, 29, 53, 93, 155, 251, 397, 613, 929, 1385, 2033, 2945, 4219, 5979, 8393, 11683, 16133, 22119, 30125, 40773, 54867, 73435, 97785, 129583, 170941, 224519, 293673, 382615, 496609, 642231, 827667, 1063073, 1361029, 1737081, 2210381, 2804485, 3548303, 4477229
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 17 2025

Keywords

Comments

In general, for m>=1, if g.f. = Sum_{k>=0} x^k * Product_{j=1..m*k} (1 + x^j)/(1 - x^j), then a(n) ~ Gamma(1/m) * exp(Pi*sqrt(n)) / (m * 2^(2 + 2/m) * Pi^(1 - 1/m) * n^((m+1)/(2*m))).

Links

Crossrefs

Cf. A035298, A385070.
Cf. A207641, A385088, A385089, A385090, A385091.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Sum[x^k*Product[(1+x^j)/(1-x^j), {j, 1, 6*k}], {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 50; p = 1; q = 1; s = 1; Do[p = Expand[p*(1 - x^(6*k))*(1 - x^(6*k - 1))*(1 - x^(6*k - 2))*(1 - x^(6*k - 3))*(1 - x^(6*k - 4))*(1 - x^(6*k - 5))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; q = Expand[q*(1 + x^(6*k))*(1 + x^(6*k - 1))*(1 + x^(6*k - 2))*(1 + x^(6*k - 3))*(1 + x^(6*k - 4))*(1 + x^(6*k - 5))]; q = Take[q, Min[nmax + 1, Exponent[q, x] + 1, Length[q]]]; s += x^k*q/p;, {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x]

Formula

a(n) ~ Gamma(1/6) * exp(Pi*sqrt(n)) / (3 * 2^(10/3) * Pi^(5/6) * n^(7/12)).

A204854 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x^k) / (1 + x^k).

Original entry on oeis.org

1, 1, -1, 1, -3, 3, -3, 3, -3, 5, -5, 5, -5, 5, -5, 5, -7, 7, -7, 7, -7, 7, -7, 7, -7, 9, -9, 9, -9, 9, -9, 9, -9, 9, -9, 9, -11, 11, -11, 11, -11, 11, -11, 11, -11, 11, -11, 11, -11, 13, -13, 13, -13, 13, -13, 13, -13, 13, -13, 13, -13, 13, -13, 13, -15, 15, -15, 15, -15, 15, -15, 15
Offset: 0

Views

Author

Paul D. Hanna, Jan 20 2012

Keywords

Examples

			G.f.: A(x) = 1 + x - x^2 + x^3 - 3*x^4 + 3*x^5 - 3*x^6 + 3*x^7 - 3*x^8 + 5*x^9 +...
where the g.f. is given by the series identities:
(0) A(x) = 1 + x*(1-x)/(1+x) + x^2*(1-x)*(1-x^2)/((1+x)*(1+x^2)) + x^3*(1-x)*(1-x^2)*(1-x^3)/((1+x)*(1+x^2)*(1+x^3)) + x^4*(1-x)*(1-x^2)*(1-x^3)*(1-x^4)/((1+x)*(1+x^2)*(1+x^3)*(1+x^4)) +...
(1) A(x) = 1 + x*(1+x^3)/(1+x) - 3*x^4*(1+x^5)/(1+x) + 5*x^9*(1+x^7)/(1+x) - 7*x^16*(1+x^9)/(1+x) + 9*x^25*(1+x^11)/(1+x) -+...
(2) A(x) = (1 + 2*x - 2*x^4 + 2*x^9 - 2*x^16 + 2*x^25 - 2*x^36 +...)/(1+x).

Links

Crossrefs

Cf. A002448 (Jacobi theta_4), A207641.

Programs

  • PARI
    {a(n)=polcoeff(1+sum(m=1,n,x^m*prod(k=1,m,(1-x^k)/(1+x^k+x*O(x^n)))),n)}

Formula

G.f.: 1 - Sum_{n>=1} (-1)^n*(2*n-1)*x^(n^2)*(1+x^(2*n+1))/(1+x).
G.f.: (1 - Sum_{n>=1} (-1)^n*2*x^(n^2)) / (1+x).

A309266 Expansion of (1 + x) * Product_{k>=1} (1 + x^k)/(1 - x^k).

Original entry on oeis.org

1, 3, 6, 12, 22, 38, 64, 104, 164, 254, 386, 576, 848, 1232, 1768, 2512, 3534, 4926, 6812, 9348, 12736, 17240, 23192, 31016, 41256, 54594, 71890, 94232, 122976, 159816, 206872, 266768, 342756, 438868, 560064, 712448, 903526, 1142478, 1440528, 1811384, 2271720, 2841800, 3546224
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 20 2019

Keywords

Crossrefs

Cf. A001650, A015128, A052816, A084376, A207641, A211971, A277643.

Programs

  • Mathematica
    nmax = 42; CoefficientList[Series[(1 + x) Product[(1 + x^k)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[PartitionsP[k] PartitionsQ[n - k], {k, 0, n}]; Table[a[n] + a[n - 1], {n, 0, 42}]

Formula

G.f.: (1 + x)/theta_4(x), where theta_4() is the Jacobi theta function.
a(n) = A015128(n) + A015128(n-1).
a(n) ~ exp(Pi*sqrt(n)) / (4*n) * (1 - (Pi/4 + 1/Pi)/sqrt(n)). - Vaclav Kotesovec, Jul 20 2019
Showing 1-8 of 8 results.

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