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.

A292167 Main diagonal of A292166.

Original entry on oeis.org

1, -1, -4, -19, -175, 5675, 184457, 43191049, 2451878210, -1954643545560, -460010080560470, -507632980962973280, -2331098560820024079917, -1447441343052056996340091, 125256629703082731968449343539, 204110605936469378207070491919090
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2017

Keywords

Links

Crossrefs

Cf. A292166.

Programs

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

Formula

a(n) = [x^n] Product_{k=1..n} (1 - k^n*x^k).

A292189 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 5, 2, 1, 1, 8, 13, 7, 3, 1, 1, 16, 35, 25, 15, 4, 1, 1, 32, 97, 91, 77, 25, 5, 1, 1, 64, 275, 337, 405, 161, 43, 6, 1, 1, 128, 793, 1267, 2177, 1069, 393, 64, 8, 1, 1, 256, 2315, 4825, 11925, 7313, 3799, 726, 120, 10
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2017

Keywords

Examples

			Square array begins:
   1, 1,  1,  1,   1, ...
   1, 1,  1,  1,   1, ...
   1, 2,  4,  8,  16, ...
   2, 5, 13, 35,  97, ...
   2, 7, 25, 91, 337, ...

Links

Crossrefs

Columns k=0..5 give A000009, A022629, A092484, A265840, A265841, A265842.
Rows 0+1, 2, 3 give A000012, A000079, A007689.
Main diagonal gives A292190.
Cf. A292166.

Programs

  • Maple
    b:= proc(n, i, k) option remember; (m->
      `if`(mn, 0, i^k*b(n-i, i-1, k)))))(i*(i+1)/2)
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 11 2017
  • Mathematica
    m = 14;
col[k_] := col[k] = Product[1 + j^k*x^j, {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

A292193 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} 1/(1 - j^k*x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 6, 5, 1, 1, 9, 14, 14, 7, 1, 1, 17, 36, 46, 25, 11, 1, 1, 33, 98, 164, 107, 56, 15, 1, 1, 65, 276, 610, 505, 352, 97, 22, 1, 1, 129, 794, 2324, 2531, 2474, 789, 198, 30, 1, 1, 257, 2316, 8986, 13225, 18580, 7273, 2314, 354, 42
Offset: 0

Views

Author

Seiichi Manyama, Sep 11 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,   1,   1, ...
   1,  1,  1,   1,   1, ...
   2,  3,  5,   9,  17, ...
   3,  6, 14,  36,  98, ...
   5, 14, 46, 164, 610, ...

Links

Crossrefs

Columns k=0..5 give A000041, A006906, A077335, A265837, A265838, A265839.
Rows 0+1, 2 give A000012, A000051.
Main diagonal gives A292194.
Cf. A292166.

Programs

  • Maple
    b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      `if`(i>n, 0, i^k*b(n-i, i, k))+b(n, i-1, k))
    end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 11 2017
  • Mathematica
    m = 12;
col[k_] := col[k] = Product[1/(1 - j^k*x^j), {j, 1, m}] + O[x]^(m+1) // CoefficientList[#, x]&;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 11 2021 *)

Formula

A(0,k) = 1 and A(n,k) = (1/n) * Sum_{j=1..n} (Sum_{d|j} d^(1+k*j/d)) * A(n-j,k) for n > 0. - Seiichi Manyama, Nov 02 2017

A294579 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals, where A(n,k) is Sum_{d|n} d^(1 + k*n/d).

Original entry on oeis.org

1, 1, 3, 1, 5, 4, 1, 9, 10, 7, 1, 17, 28, 25, 6, 1, 33, 82, 97, 26, 12, 1, 65, 244, 385, 126, 80, 8, 1, 129, 730, 1537, 626, 588, 50, 15, 1, 257, 2188, 6145, 3126, 4508, 344, 161, 13, 1, 513, 6562, 24577, 15626, 35652, 2402, 2049, 163, 18
Offset: 1

Views

Author

Seiichi Manyama, Nov 02 2017

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,    1, ...
  3,  5,   9,  17,   33, ...
  4, 10,  28,  82,  244, ...
  7, 25,  97, 385, 1537, ...
  6, 26, 126, 626, 3126, ...

Links

Crossrefs

Columns k=0..2 give A000203, A078308, A294567.
Rows k=0..1 give A000012, A000051(n+1).
Cf. A292166, A292193.

Formula

L.g.f. of column k: -log(Product_{j>=1} (1 - j^k * x^j)). - Seiichi Manyama, Jun 02 2019
G.f. of column k: Sum_{j>0} j^(k+1) * x^j / (1 - j^k * x^j). - Seiichi Manyama, Jan 14 2023

A292164 Expansion of Product_{k>=1} (1 - k^2*x^k).

Original entry on oeis.org

1, -1, -4, -5, -7, 27, 17, 167, 110, -42, 10, -706, -4001, -3915, 3079, -18640, 9869, 21403, 130565, 107250, -15661, 420664, 599540, -161785, -1232833, -5836888, -5129796, 6516714, -29068180, -14953045, -41490510, 20261320, 30395771, 441235155, 205289550
Offset: 0

Views

Author

Seiichi Manyama, Sep 10 2017

Keywords

Links

Crossrefs

Column k=2 of A292166.
Cf. A022629, A022661, A077335, A092484.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, i^2*b(n-i, i))))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      -add(b(n-i$2)*a(i$2), i=0..n-1))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 10 2017
  • Mathematica
    b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
    b[n, i - 1] + If[i > n, 0, i^2*b[n - i, i]]]];
a[n_] := a[n] = If[n == 0, 1,
    -Sum[b[n - i, n - i]*a[i], {i, 0, n - 1}]];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 04 2024, after Alois P. Heinz *)
  • PARI
    N=66; x='x+O('x^N); Vec(prod(n=1, N, 1-n^2*x^n))

Formula

Convolution inverse of A077335.
G.f.: exp(-Sum_{k>=1} Sum_{j>=1} j^(2*k)*x^(j*k)/k). - Ilya Gutkovskiy, Jun 18 2018

A294580 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - j^k*x^j)^j.

Original entry on oeis.org

1, 1, -1, 1, -1, -2, 1, -1, -4, -1, 1, -1, -8, -5, 0, 1, -1, -16, -19, -3, 4, 1, -1, -32, -65, -21, 23, 4, 1, -1, -64, -211, -111, 139, 44, 7, 1, -1, -128, -665, -525, 863, 448, 104, 3, 1, -1, -256, -2059, -2343, 5419, 4316, 1414, 70, -2, 1, -1, -512, -6305, -10101, 34103, 40024, 18164, 1206, -93, -9
Offset: 0

Views

Author

Seiichi Manyama, Nov 02 2017

Keywords

Examples

			Square array begins:
    1,  1,   1,    1,    1, ...
   -1, -1,  -1,   -1,   -1, ...
   -2, -4,  -8,  -16,  -32, ...
   -1, -5, -19,  -65, -211, ...
    0, -3, -21, -111, -525, ...

Links

Crossrefs

Columns k=0..2 give A073592, A266964, A294581.
Rows n=0..3 give A000012, (-1)*A000012, (-1)*A000079(n+1), (-1)*A001047(n+1).
Cf. A292166, A294582.

Formula

A(0,k) = 1 and A(n,k) = -(1/n) * Sum_{j=1..n} (Sum_{d|j} d^(2+k*j/d)) * A(n-j,k) for n > 0.
Showing 1-6 of 6 results.

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