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.

A292892 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (exp(x) - 1)).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 5, 1, 0, 0, 3, 15, 1, 0, 0, 6, 16, 52, 1, 0, 0, 0, 12, 65, 203, 1, 0, 0, 0, 24, 20, 336, 877, 1, 0, 0, 0, 0, 60, 390, 1897, 4140, 1, 0, 0, 0, 0, 120, 120, 2562, 11824, 21147, 1, 0, 0, 0, 0, 0, 360, 210, 11816, 80145, 115975, 1, 0, 0, 0, 0, 0, 720, 840, 20496, 105912, 586000, 678570
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2017

Keywords

Examples

			Square array begins:
   1,  1,  1,  1,   1, ...
   1,  0,  0,  0,   0, ...
   2,  2,  0,  0,   0, ...
   5,  3,  6,  0,   0, ...
  15, 16, 12, 24,   0, ...
  52, 65, 20, 60, 120, ...

Links

Crossrefs

Columns k=0..3 give A000110, A052506, A240989, A292891.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
Cf. A292894, A355607.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

A355609 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 6, 20, 120, 1, 0, 0, 0, 12, 90, 720, 1, 0, 0, 0, 24, 40, 594, 5040, 1, 0, 0, 0, 0, 60, 540, 4200, 40320, 1, 0, 0, 0, 0, 120, 240, 3528, 34544, 362880, 1, 0, 0, 0, 0, 0, 360, 1260, 25200, 316008, 3628800, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, 263520, 3207240, 39916800
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Examples

			Square array begins:
    1,   1,   1,   1,   1,   1, 1, ...
    1,   0,   0,   0,   0,   0, 0, ...
    2,   2,   0,   0,   0,   0, 0, ...
    6,   3,   6,   0,   0,   0, 0, ...
   24,  20,  12,  24,   0,   0, 0, ...
  120,  90,  40,  60, 120,   0, 0, ...
  720, 594, 540, 240, 360, 720, 0, ...

Links

Crossrefs

Columns k=0..4 give A000142, A066166, A353228, A353229, A354624.
Cf. A355607, A355610.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), abs(stirling(n-k*j, j, 1))/(n-k*j)!);

Formula

T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(n-k*j)!.

A292894 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^k * (1 - exp(x))).

Original entry on oeis.org

1, 1, -1, 1, 0, 0, 1, 0, -2, 1, 1, 0, 0, -3, 1, 1, 0, 0, -6, 8, -2, 1, 0, 0, 0, -12, 55, -9, 1, 0, 0, 0, -24, -20, 84, -9, 1, 0, 0, 0, 0, -60, 330, -637, 50, 1, 0, 0, 0, 0, -120, -120, 2478, -4992, 267, 1, 0, 0, 0, 0, 0, -360, -210, 11704, -10593, 413, 1, 0, 0, 0, 0, 0, -720, -840, 19824, -15192, 92060, -2180
Offset: 0

Views

Author

Seiichi Manyama, Sep 26 2017

Keywords

Examples

			Square array begins:
   1,  1,   1,   1,    1, ...
  -1,  0,   0,   0,    0, ...
   0, -2,   0,   0,    0, ...
   1, -3,  -6,   0,    0, ...
   1,  8, -12, -24,    0, ...
  -2, 55, -20, -60, -120, ...

Links

Crossrefs

Columns k=0..2 give A000587, A292893, A292951.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
Cf. A292892, A355607.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), (-1)^j*stirling(n-k*j, j, 2)/(n-k*j)!); \\ Seiichi Manyama, Jul 09 2022

Formula

From Seiichi Manyama, Jul 09 2022: (Start)
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} (-1)^j * Stirling2(n-k*j,j)/(n-k*j)!.
T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} j/(j-k)! * T(n-j,k)/(n-j)! for n > 0. (End)

A355619 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k/k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 3, 20, 0, 1, 0, 0, 0, -6, -90, 0, 1, 0, 0, 0, 4, 20, 594, 0, 1, 0, 0, 0, 0, -10, 0, -4200, 0, 1, 0, 0, 0, 0, 5, 40, -126, 34544, 0, 1, 0, 0, 0, 0, 0, -15, -210, 1260, -316008, 0, 1, 0, 0, 0, 0, 0, 6, 70, 1904, -4320, 3207240, 0
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 2022

Keywords

Examples

			Square array begins:
  1,   1,  1,   1,   1, 1, 1, ...
  1,   0,  0,   0,   0, 0, 0, ...
  0,   2,  0,   0,   0, 0, 0, ...
  0,  -3,  3,   0,   0, 0, 0, ...
  0,  20, -6,   4,   0, 0, 0, ...
  0, -90, 20, -10,   5, 0, 0, ...
  0, 594,  0,  40, -15, 6, 0, ...

Links

Crossrefs

Columns k=1..4 give A007113, A355605, (-1)^n * A351493(n), A355603.
Cf. A355607, A355610.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(k!^j*(n-k*j)!));

Formula

T(0,k) = 1 and T(n,k) = -(n-1)!/k! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(k!^j * (n-k*j)!).

A354625 Expansion of e.g.f. (1 + x)^(x^4).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, -360, 1680, -10080, 72576, 1209600, -14256000, 159667200, -1902700800, 24458353920, -120860812800, -193037644800, 23690780467200, -646842994237440, 14916006359654400, -230812655044608000, 3182953434006528000, -37667817509059584000
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Crossrefs

Column k=4 of A355607.
Cf. A007121, A354624.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1+x)^x^4))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4*log(1+x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-(i-1)!*sum(j=5, i,(-1)^j*j/(j-4)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\5, stirling(n-4*k, k, 1)/(n-4*k)!);

Formula

a(0) = 1; a(n) = -(n-1)! * Sum_{k=5..n} (-1)^k * k/(k-4) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/5)} Stirling1(n-4*k,k)/(n-4*k)!.
Showing 1-5 of 5 results.

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