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.

A084376 G.f.: (1+x)/Product_{m>0} (1 - x^m).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 26, 37, 52, 72, 98, 133, 178, 236, 311, 407, 528, 682, 875, 1117, 1419, 1794, 2257, 2830, 3533, 4394, 5446, 6728, 8283, 10169, 12446, 15191, 18492, 22453, 27193, 32860, 39614, 47652, 57200, 68523, 81921, 97757, 116435, 138436
Offset: 0

Views

Author

Vladeta Jovovic, Jun 23 2003

Keywords

Links

Crossrefs

Cf. A052816.

Programs

  • Maple
    seq(numbpart(k)+numbpart(k+1), k=0..43); # Zerinvary Lajos, Jun 06 2007
  • Mathematica
    Table[PartitionsP[n] + PartitionsP[n - 1], {n, 0, 44}] (* Robert Price, May 18 2020 *)

Formula

a(n) = A000041(n) + A000041(n-1), n>0.
a(n) ~ exp(sqrt(2*n/3)*Pi)/(2*sqrt(3)*n) * (1 - (sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n) + (13/16 + 313*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

A329289 Expansion of g.f. (1 + x) * (1 + x^2) * Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 82, 99, 119, 143, 170, 202, 240, 283, 333, 391, 457, 533, 621, 721, 835, 966, 1114, 1282, 1474, 1690, 1935, 2213, 2525, 2877, 3274, 3719, 4219, 4781, 5409, 6112, 6900, 7778, 8758, 9852, 11068, 12422
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 07 2020

Keywords

Comments

Number of partitions of n into distinct parts if there are two types of 1's and two types of 2's.

Crossrefs

Cf. A000009, A000097, A022567, A036469, A052816, A096914, A329384.

Programs

  • Maple
    series(1/2 * add( x^((n-2)*(n-3)/2) / mul(1 - x^k, k = 1..n), n = 0..12), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Feb 03 2025
  • Mathematica
    nmax = 50; CoefficientList[Series[(1 + x) (1 + x^2) Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) If[d < 3, 2, 1] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 50}]

Formula

a(n) = A036469(n) - A036469(n-4).
a(n) ~ exp(Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 11 2020
G.f.: A(x) = 1/2 * Sum_{n >= 0} x^((n-2)*(n-3)/2) / (Product_{k = 1..n} 1 - x^k). - Peter Bala, Feb 03 2025

A309267 Expansion of (1 + x) * Product_{k>=1} 1/(1 - x^k)^k.

Original entry on oeis.org

1, 2, 4, 9, 19, 37, 72, 134, 246, 442, 782, 1359, 2338, 3964, 6652, 11046, 18176, 29631, 47935, 76931, 122608, 194072, 305269, 477258, 741977, 1147227, 1764778, 2701403, 4115892, 6242846, 9428575, 14181272, 21245738, 31708402, 47150928, 69867001, 103176007, 151864745, 222821779
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 20 2019

Keywords

Links

Crossrefs

Cf. A000219, A052816, A084376, A091360, A191659, A277963.

Programs

  • Maple
    G:= (1+x)/mul((1-x^k)^k,k=1..100):
S:= series(G,x,101):
seq(coeff(S,x,j),j=0..100); # Robert Israel, Dec 01 2020
  • Mathematica
    nmax = 38; CoefficientList[Series[(1 + x) Product[1/(1 - x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[DivisorSigma[2, k] a[n - k], {k, 1, n}]/n; Table[a[n] + a[n - 1], {n, 0, 38}]

Formula

a(n) = A000219(n) + A000219(n-1).
a(n) ~ Zeta(3)^(7/36) * 2^(25/36) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Jul 20 2019

A329384 G.f.: (1 + x) * (1 + x^2) * (1 + x^3) * Product_{k>=1} (1 + x^k).

Original entry on oeis.org

1, 2, 3, 6, 8, 11, 16, 20, 26, 34, 43, 54, 68, 84, 103, 127, 154, 186, 225, 269, 321, 383, 453, 535, 631, 740, 866, 1012, 1178, 1368, 1587, 1835, 2117, 2440, 2804, 3217, 3687, 4215, 4812, 5487, 6244, 7096, 8055, 9128, 10331, 11681, 13187, 14870, 16752, 18846, 21180
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 07 2020

Keywords

Comments

Number of partitions of n into distinct parts if there are two types of 1's, two types of 2's and two types of 3's.

Crossrefs

Cf. A000009, A000098, A022567, A036469, A052816, A329289.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[(1 + x) (1 + x^2) (1 + x^3) Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) If[d < 4, 2, 1] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 50}]

Formula

a(n) = A036469(n) + A036469(n-3) - A036469(n-4) - A036469(n-7).
a(n) ~ 2*exp(Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 11 2020

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 1-5 of 5 results.

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