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 1-7 of 7 results.

A293072 G.f.: Product_{m>0} (1-x^m+2!*x^(2*m)).

Original entry on oeis.org

1, -1, 1, 0, 0, -3, 4, -3, 4, -2, 0, -6, 9, -10, 10, -5, 10, -16, 20, -28, 22, -18, 19, -32, 54, -50, 45, -38, 44, -78, 106, -118, 96, -98, 110, -129, 192, -216, 204, -182, 213, -286, 368, -412, 366, -362, 394, -524, 676, -714, 680, -641, 742, -936, 1170
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2017

Keywords

Links

Crossrefs

Column k=2 of A293071.
Cf. A293204, A293304.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!*(-1)^j, j=0..min(2, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 04 2017
  • Mathematica
    nmax = 100; CoefficientList[Series[Product[1 - x^k + 2*x^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2017 *)
  • PARI
    Vec(prod(m=1, 80, 1-x^m+2*x^(2*m)) + O(x^80)) \\ Michel Marcus, Oct 04 2017

A293305 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} (1 + Sum_{j=1..k} (-1)^j*j*x^(j*i)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, -1, 0, 1, -1, 1, 0, 0, 1, -1, 1, 0, 0, 0, 1, -1, 1, -3, 0, 1, 0, 1, -1, 1, -3, 0, -3, 0, 0, 1, -1, 1, -3, 4, 0, 4, 1, 0, 1, -1, 1, -3, 4, 0, 4, -3, 0, 0, 1, -1, 1, -3, 4, -5, 0, -3, 4, 0, 0, 1, -1, 1, -3, 4, -5, 0, -7, -2, -2, 0, 0, 1, -1, 1
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0, -1,  1,  1,  1, ...
   0,  0,  0, -3, -3, ...
   0,  0,  0,  0,  4, ...
   0,  1, -3,  0,  0, ...

Links

Crossrefs

Columns k=0..2 give A000007, A010815, A293072.
Rows n=0 gives A000012.
Main diagonal gives A293306.
Cf. A293071, A293307.

Programs

  • Mathematica
    nmax = 12;
col[k_] := col[k] = Product[1+Sum[(-1)^j*j*x^(i*j), {j, 1, k}], {i, 1, 2 nmax}] + O[x]^(2 nmax) // CoefficientList[#, x]&;
A[n_, k_] := If[n == 0, 1, If[k == 0, 0, col[k][[n+1]]]];
Table[A[n-k, k], {n, 0, nmax}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 21 2021 *)

A293236 G.f.: Product_{i>0} Sum_{j>=0} (-1)^j*j!*x^(j*i).

Original entry on oeis.org

1, -1, 1, -6, 24, -117, 700, -4947, 39760, -358682, 3594084, -39598866, 475774299, -6191078998, 86742689434, -1301964957707, 20842304366686, -354473010919852, 6382843971860354, -121311619900081996, 2426875912883720386, -50976050128395861672
Offset: 0

Views

Author

Seiichi Manyama, Oct 03 2017

Keywords

Links

Crossrefs

Cf. A293071.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!*(-1)^j, j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Oct 04 2017
  • Mathematica
    m = 22;
CoefficientList[Product[Sum[(-1)^j j! x^(i j), {j, 0, m}], {i, 1, m}] + O[x]^m, x] (* Jean-François Alcover, Nov 15 2020 *)

Formula

a(n) ~ (-1)^n * n! * (1 - 1/n^2 + 3/n^4 + 12/n^5 + 35/n^6 + 61/n^7 - 153/n^8 - 2197/n^9 - 11330/n^10), for coefficients see A293267. - Vaclav Kotesovec, Oct 04 2017

A293293 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of g.f. Product_{i>0} 1/(Sum_{j=0..k} (-1)^j*j!*x^(j*i)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, -1, 5, 0, 1, 1, 0, 5, -1, 7, 0, 1, 1, 0, 5, 11, 3, 11, 0, 1, 1, 0, 5, -13, 3, 3, 15, 0, 1, 1, 0, 5, -13, -45, 15, -1, 22, 0, 1, 1, 0, 5, -13, 75, 15, 77, -8, 30, 0, 1, 1, 0, 5, -13, 75, 255, -91, 76, -8, 42, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 05 2017

Keywords

Examples

			Square array begins:
   1, 1,  1,  1,   1, ...
   0, 1,  1,  1,   1, ...
   0, 2,  0,  0,   0, ...
   0, 3, -1,  5,   5, ...
   0, 5, -1, 11, -13, ...
   0, 7,  3,  3, -45, ...

Links

Crossrefs

Columns k=0..2 give A000007, A000041, A293294.
Rows n=0..1 give A000012, A057427.
Main diagonal gives A293259.
Cf. A293071, A293285.

A293255 G.f.: Product_{m>0} (1 - x^m + 2!*x^(2*m) - 3!*x^(3*m)).

Original entry on oeis.org

1, -1, 1, -6, 0, 3, 4, -3, -8, 22, 12, 6, 3, -22, -62, 61, -2, 68, -166, 68, -14, -216, -53, 196, -12, 304, -15, -32, -322, 24, 274, -112, -294, 832, 638, -213, 918, -474, -570, -854, -1227, -1018, 3476, -2038, 588, -1382, 3520, -578, -248, -1314, -292, -8825
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2017

Keywords

Links

Crossrefs

Column k=3 of A293071.
Cf. A289485.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!*(-1)^j, j=0..min(3, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 04 2017

A293256 G.f.: Product_{m>0} (1 - x^m + 2!*x^(2*m) - 3!*x^(3*m) + 4!*x^(4*m)).

Original entry on oeis.org

1, -1, 1, -6, 24, 3, -20, -27, 40, -2, -36, -186, 579, -118, 274, -347, 46, -892, 74, -76, 3106, -3192, -1013, -4844, 12276, -752, 2145, -656, 11198, -11160, -2246, -38752, 32106, -24272, 42182, -7533, 36054, -78906, -4482, -164462, 381837, -174346, 21356, -114910
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2017

Keywords

Links

Crossrefs

Column k=4 of A293071.
Cf. A289486.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!*(-1)^j, j=0..min(4, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 04 2017
  • PARI
    Vec(prod(m=1, 50, 1+sum(k=1, 4, (-1)^k*k!*x^(k*m))) + O(x^50)) \\ Michel Marcus, Oct 04 2017

A293257 G.f.: Product_{m>0} (1 - x^m + 2!*x^(2*m) - 3!*x^(3*m) + 4!*x^(4*m) - 5!*x^(5*m)).

Original entry on oeis.org

1, -1, 1, -6, 24, -117, -20, 93, 160, -122, -156, 414, 459, -2038, -2486, 13333, 1246, 2468, -8446, -13036, -4814, 7848, -11453, 50116, -46764, -1952, -164895, 214144, 249038, 396600, -1598966, 254048, -465174, -559712, 726422, 747, 1067574, 2897334, 165438
Offset: 0

Views

Author

Seiichi Manyama, Oct 04 2017

Keywords

Links

Crossrefs

Column k=5 of A293071.
Cf. A293250.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*j!*(-1)^j, j=0..min(5, n/i))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 04 2017
  • PARI
    Vec(prod(m=1, 50, 1+sum(k=1, 5, (-1)^k*k!*x^(k*m))) + O(x^50)) \\ Michel Marcus, Oct 04 2017
Showing 1-7 of 7 results.

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