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 1-6 of 6 results.

A317665 Expansion of 1/Sum_{k>=0} x^(k^2).

Original entry on oeis.org

1, -1, 1, -1, 0, 1, -2, 3, -3, 1, 2, -6, 10, -11, 8, 0, -14, 29, -39, 38, -18, -22, 74, -123, 144, -110, 6, 161, -352, 491, -484, 251, 235, -896, 1528, -1825, 1452, -191, -1892, 4317, -6164, 6243, -3488, -2482, 10788, -18957, 23140, -19085, 3858, 22025, -52833, 77224, -80198, 47899
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 08 2018

Keywords

Comments

Convolution inverse of A010052.

Examples

			G.f. = 1 - x + x^2 - x^3 + x^5 - 2*x^6 + 3*x^7 - 3*x^8 + x^9 + 2*x^10 - 6*x^11 + ...

Links

Crossrefs

Cf. A004402, A006456, A010052, A106507, A323633, A337165.

Programs

  • Maple
    a:=series(1/add(x^(k^2),k=0..100),x=0,54): seq(coeff(a,x,n),n=0..53); # Paolo P. Lava, Apr 02 2019
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(`if`(issqr(n-j), a(j), 0), j=0..n-1))
    end:
seq(a(n), n=0..53);  # Alois P. Heinz, Jul 26 2025
  • Mathematica
    nmax = 53; CoefficientList[Series[1/Sum[x^k^2, {k, 0, nmax}], {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[2/(1 + QPochhammer[x^2]^5/(QPochhammer[x] QPochhammer[x^4])^2), {x, 0, nmax}], x]
nmax = 53; CoefficientList[Series[2/(1 + EllipticTheta[3, 0, q]), {q, 0, nmax}], q]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[IntegerQ[Sqrt[k]]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 53}]
  • PARI
    seq(n)=Vec(1/(sum(k=0, sqrtint(n), x^(k^2)) + O(x*x^n))) \\ Andrew Howroyd, Aug 08 2018
  • PARI
    a(n) = if(n==0, 1, -sum(k=1, n, issquare(k)*a(n-k))); \\ Seiichi Manyama, Mar 19 2022

Formula

G.f.: 2/(1 + theta_3(q)), where theta_3() is the Jacobi theta function.
a(n) = Sum_{k=0..n} (-1)^k * A337165(n,k).
a(0) = 1; a(n) = -Sum_{k=1..n} A010052(k) * a(n-k). - Seiichi Manyama, Mar 19 2022

A363779 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of 1/(Sum_{j>=0} x^(j^3))^k.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 1, 0, 1, -3, 3, -1, 0, 1, -4, 6, -4, 1, 0, 1, -5, 10, -10, 5, -1, 0, 1, -6, 15, -20, 15, -6, 1, 0, 1, -7, 21, -35, 35, -21, 7, -1, 0, 1, -8, 28, -56, 70, -56, 28, -8, 0, 0, 1, -9, 36, -84, 126, -126, 84, -36, 7, 1, 0, 1, -10, 45, -120, 210, -252, 210, -120, 42, -4, -2, 0
Offset: 0

Views

Author

Seiichi Manyama, Jun 21 2023

Keywords

Examples

			Square array begins:
  1,  1,  1,   1,   1,    1,    1, ...
  0, -1, -2,  -3,  -4,   -5,   -6, ...
  0,  1,  3,   6,  10,   15,   21, ...
  0, -1, -4, -10, -20,  -35,  -56, ...
  0,  1,  5,  15,  35,   70,  126, ...
  0, -1, -6, -21, -56, -126, -252, ...
  0,  1,  7,  28,  84,  210,  462, ...

Crossrefs

Columns k=0..3 give A000007, A323633, A363776, A363777.
Main diagonal gives A363781.
Cf. A290054, A363778, A363783.

Formula

T(0,k) = 1; T(n,k) = -(k/n) * Sum_{j=1..n} A363783(j) * T(n-j,k).

A352529 Expansion of 1/Sum_{k>=0} x^(k^4).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -15, 14, -12, 9, -5, 0, 6, -13, 21, -30, 40, -51, 63, -76, 90, -105, 120, -134, 146, -155, 160, -160, 154, -141, 120, -90, 50, 1, -64, 140, -230, 335, -455, 589, -735, 890, -1050, 1210, -1364, 1505
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2022

Keywords

Links

Crossrefs

Cf. A000583, A317665, A323633, A352530.

Programs

  • PARI
    my(N=99, x='x+O('x^N)); Vec(1/sum(k=0, N^(1/4), x^k^4))
  • PARI
    a(n) = if(n==0, 1, -sum(k=1, n, ispower(k, 4)*a(n-k)));

A352530 Expansion of 1/Sum_{k>=0} x^(k^5).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, -18, 19, -20, 21, -22, 23, -24, 25, -26, 27, -28, 29, -30, 31, -31, 30, -28, 25, -21, 16, -10, 3, 5, -14, 24, -35, 47, -60, 74, -89, 105, -122
Offset: 0

Views

Author

Seiichi Manyama, Mar 19 2022

Keywords

Links

Crossrefs

Cf. A000584, A317665, A323633, A352529.

Programs

  • PARI
    my(N=99, x='x+O('x^N)); Vec(1/sum(k=0, N^(1/5), x^k^5))
  • PARI
    a(n) = if(n==0, 1, -sum(k=1, n, ispower(k, 5)*a(n-k)));

A339420 Number of compositions (ordered partitions) of n into an even number of cubes.

Original entry on oeis.org

1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 4, 1, 6, 1, 8, 2, 10, 7, 12, 16, 14, 29, 16, 46, 22, 67, 40, 94, 78, 125, 144, 161, 246, 214, 394, 312, 602, 499, 878, 835, 1236, 1396, 1722, 2286, 2446, 3637, 3614, 5598, 5560, 8358, 8782, 12226, 14014, 17776, 22278, 26056, 34924
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(11) = 4 because we have [8, 1, 1, 1], [1, 8, 1, 1], [1, 1, 8, 1] and [1, 1, 1, 8].

Links

Crossrefs

Cf. A000578, A023358, A034008, A323633, A339368, A339418, A339421.

Programs

  • Maple
    b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) + 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) + 1 / Sum_{k>=0} x^(k^3)).
a(n) = (A023358(n) + A323633(n)) / 2.
a(n) = Sum_{k=0..n} A023358(k) * A323633(n-k).

A339421 Number of compositions (ordered partitions) of n into an odd number of cubes.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 3, 1, 5, 1, 7, 1, 9, 4, 11, 11, 13, 22, 15, 37, 18, 56, 29, 80, 56, 109, 107, 142, 190, 184, 313, 255, 490, 391, 731, 644, 1045, 1082, 1458, 1792, 2044, 2895, 2957, 4531, 4463, 6863, 6972, 10126, 11090, 14739, 17691, 21484, 27954, 31741
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(10) = 3 because we have [8, 1, 1], [1, 8, 1] and [1, 1, 8].

Links

Crossrefs

Cf. A000578, A023358, A166444, A323633, A339369, A339419, A339420.

Programs

  • Maple
    b:= proc(n, t) option remember; local r, f, g;
      if n=0 then t else r, f, g:=$0..2; while f<=n
      do r, f, g:= r+b(n-f, 1-t), f+3*g*(g-1)+1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 57; CoefficientList[Series[(1/2) (1/(1 - Sum[x^(k^3), {k, 1, Floor[nmax^(1/3)] + 1}]) - 1/Sum[x^(k^3), {k, 0, Floor[nmax^(1/3)] + 1}]), {x, 0, nmax}], x]

Formula

G.f.: (1/2) * (1 / (1 - Sum_{k>=1} x^(k^3)) - 1 / Sum_{k>=0} x^(k^3)).
a(n) = (A023358(n) - A323633(n)) / 2.
a(n) = -Sum_{k=0..n-1} A023358(k) * A323633(n-k).
Showing 1-6 of 6 results.

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