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 1-10 of 12 results. Next

A337165 Number T(n,k) of compositions of n into k nonzero squares; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 5, 0, 0, 1, 0, 1, 0, 3, 0, 0, 6, 0, 0, 1, 0, 0, 2, 0, 6, 0, 0, 7, 0, 0, 1, 0, 0, 0, 3, 0, 10, 0, 0, 8, 0, 0, 1, 0, 0, 0, 1, 4, 0, 15, 0, 0, 9, 0, 0, 1
Offset: 0

Views

Author

Alois P. Heinz, Feb 03 2021

Keywords

Examples

			Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 0, 1;
  0, 1, 0, 0, 1;
  0, 0, 2, 0, 0,  1;
  0, 0, 0, 3, 0,  0,  1;
  0, 0, 0, 0, 4,  0,  0, 1;
  0, 0, 1, 0, 0,  5,  0, 0, 1;
  0, 1, 0, 3, 0,  0,  6, 0, 0, 1;
  0, 0, 2, 0, 6,  0,  0, 7, 0, 0, 1;
  0, 0, 0, 3, 0, 10,  0, 0, 8, 0, 0, 1;
  0, 0, 0, 1, 4,  0, 15, 0, 0, 9, 0, 0, 1;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A010052, A063725, A063691, A063730, A340481, A340905, A340906, A340915, A340946, A340947.
Row sums give A006456.
T(2n,n) gives A338464.
Main diagonal gives A000012.
Cf. A000290, A281704, A317665, A341040.

Programs

  • Maple
    b:= proc(n) option remember; `if`(n=0, 1, add((s->
     `if`(s>n, 0, expand(x*b(n-s))))(j^2), j=1..isqrt(n)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
seq(T(n), n=0..14);
  • Mathematica
    b[n_] := b[n] = If[n == 0, 1, Sum[With[{s = j^2},
     If[s>n, 0, Expand[x*b[n - s]]]], {j, 1, Sqrt[n]}]];
T[n_] := CoefficientList[b[n], x];
T /@ Range[0, 14] // Flatten (* Jean-François Alcover, Feb 07 2021, after Alois P. Heinz *)

Formula

G.f. of column k: (Sum_{j>=1} x^(j^2))^k.
Sum_{k=0..n} k * T(n,k) = A281704(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A317665(n).

A323633 Expansion of 1/Sum_{k>=0} x^(k^3).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -6, 7, -7, 6, -4, 1, 3, -8, 14, -21, 28, -34, 38, -40, 38, -31, 18, 2, -29, 62, -99, 139, -178, 211, -232, 234, -210, 154, -62, -70, 242, -449, 680, -917, 1135, -1303, 1386, -1344, 1136, -725, 85, 794, -1898, 3183, -4571
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 21 2019

Keywords

Comments

Convolution inverse of A010057.

Links

Crossrefs

Cf. A010057, A023358, A317665.

Programs

  • Maple
    a:=series(1/add(x^(k^3),k=0..100),x=0,59): seq(coeff(a,x,n),n=0..58); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 58; CoefficientList[Series[1/Sum[x^k^3, {k, 0, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = -Sum[Boole[IntegerQ[k^(1/3)]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 58}]
  • PARI
    my(N=66, x='x+O('x^N)); Vec(1/sum(k=0, N^(1/3), x^k^3)) \\ Seiichi Manyama, Mar 19 2022
  • PARI
    a(n) = if(n==0, 1, -sum(k=1, n, ispower(k, 3)*a(n-k))); \\ Seiichi Manyama, Mar 19 2022

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A010057(k) * a(n-k). - Seiichi Manyama, Mar 19 2022

A339419 Number of compositions (ordered partitions) of n into an odd number of squares.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 3, 1, 5, 5, 7, 14, 10, 27, 27, 44, 69, 73, 144, 158, 260, 366, 466, 775, 940, 1490, 2031, 2803, 4264, 5551, 8460, 11525, 16399, 23864, 32435, 47981, 66005, 94701, 135072, 187999, 272678, 379095, 543626, 769490, 1083788, 1553661, 2177681, 3113333
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(9) = 5 because we have [9], [4, 4, 1], [4, 1, 4], [1, 4, 4] and [1, 1, 1, 1, 1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000290, A006456, A166444, A317665, A339365, A339417, A339418.

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+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 47; CoefficientList[Series[1/(3 - EllipticTheta[3, 0, x]) - 1/(1 + EllipticTheta[3, 0, x]), {x, 0, nmax}], x]

Formula

G.f.: 1 / (3 - theta_3(x)) - 1 / (1 + theta_3(x)), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) - A317665(n)) / 2.
a(n) = -Sum_{k=0..n-1} A006456(k) * A317665(n-k).

A363778 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^2))^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, 0, 0, 1, -5, 10, -10, 3, 1, 0, 1, -6, 15, -20, 12, 0, -2, 0, 1, -7, 21, -35, 31, -9, -5, 3, 0, 1, -8, 28, -56, 65, -36, -2, 12, -3, 0, 1, -9, 36, -84, 120, -96, 24, 24, -18, 1, 0, 1, -10, 45, -120, 203, -210, 105, 20, -54, 18, 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,  0,  3,  12,  31,  65,  120, ...
  0,  1,  0,  -9, -36, -96, -210, ...
  0, -2, -5,  -2,  24, 105,  294, ...

Crossrefs

Columns k=0..3 give A000007, A317665, A363774, A363775.
Main diagonal gives A363780.
Cf. A045847, A162552, A363779.

Formula

T(0,k) = 1; T(n,k) = -(k/n) * Sum_{j=1..n} A162552(j) * T(n-j,k).

A308806 Expansion of 1 / Sum_{k>=0} (-x)^(k*(3*k - 1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 23, 30, 40, 52, 67, 86, 111, 145, 188, 243, 314, 406, 527, 683, 883, 1141, 1475, 1910, 2474, 3201, 4140, 5355, 6929, 8968, 11603, 15009, 19416, 25121, 32507, 42060, 54413, 70393, 91071, 117831, 152453, 197238, 255175, 330137, 427130, 552620
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 25 2019

Keywords

Crossrefs

Cf. A000041, A000326, A006950, A181324, A208061, A317665.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/Sum[(-x)^(k (3 k - 1)/2), {k, 0, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 / Sum_{k>=0} (-x)^A000326(k).

A339418 Number of compositions (ordered partitions) of n into an even number of squares.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 1, 4, 2, 6, 9, 8, 20, 16, 35, 44, 55, 102, 105, 196, 242, 344, 540, 652, 1084, 1380, 2037, 2964, 3912, 6042, 7976, 11776, 16634, 22968, 33963, 46156, 67457, 94510, 133180, 192316, 266514, 385338, 540138, 767008, 1094576, 1534704, 2200821, 3094248
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 03 2020

Keywords

Examples

			a(9) = 6 because we have [4, 1, 1, 1, 1, 1], [1, 4, 1, 1, 1, 1], [1, 1, 4, 1, 1, 1], [1, 1, 1, 4, 1, 1], [1, 1, 1, 1, 4, 1] and [1, 1, 1, 1, 1, 4].

Links

Crossrefs

Cf. A000290, A006456, A034008, A317665, A339364, A339416, A339419.

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+2*g-1, g+1 od; r fi
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 03 2020
  • Mathematica
    nmax = 47; CoefficientList[Series[4/(3 + 2 EllipticTheta[3, 0, x] - EllipticTheta[3, 0, x]^2), {x, 0, nmax}], x]

Formula

G.f.: 4 / (3 + 2 * theta_3(x) - theta_3(x)^2), where theta_3() is the Jacobi theta function.
a(n) = (A006456(n) + A317665(n)) / 2.
a(n) = Sum_{k=0..n} A006456(k) * A317665(n-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)));

A361979 Expansion of 1 / Sum_{k>=0} x^(k*(2*k - 1)).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 0, 1, -2, 3, -4, 5, -5, 4, -2, -2, 7, -13, 19, -24, 27, -25, 17, -2, -20, 48, -80, 110, -132, 137, -116, 62, 30, -158, 314, -479, 622, -704, 680, -507, 150, 405, -1135, 1973, -2797, 3432, -3662, 3250, -1983, -280, 3540, -7592, 11977, -15953
Offset: 0

Views

Author

Ilya Gutkovskiy, May 25 2023

Keywords

Crossrefs

Cf. A000384, A006950, A106507, A308806, A317665, A322798, A363149, A363275.

Programs

  • Mathematica
    nmax = 53; CoefficientList[Series[1/Sum[x^(k (2 k - 1)), {k, 0, nmax}], {x, 0, nmax}], x]

A363149 Expansion of 1 / Sum_{k>=0} x^(k*(5*k - 3)/2).

Original entry on oeis.org

1, -1, 1, -1, 1, -1, 1, -2, 3, -4, 5, -6, 7, -8, 10, -13, 17, -22, 27, -33, 40, -49, 61, -77, 98, -123, 153, -189, 233, -288, 358, -448, 561, -701, 872, -1082, 1342, -1666, 2073, -2584, 3223, -4016, 4997, -6212, 7720, -9598, 11942, -14869, 18517, -23053, 28687
Offset: 0

Views

Author

Ilya Gutkovskiy, May 25 2023

Keywords

Crossrefs

Cf. A000566, A106507, A308806, A317665, A322799, A361979, A363275.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[1/Sum[x^(k (5 k - 3)/2), {k, 0, nmax}], {x, 0, nmax}], x]
Showing 1-10 of 12 results. Next

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