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.

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)!.

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)!).

A355652 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k/k! * log(1 - x)).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 14, 1, 0, 0, 3, 88, 1, 0, 0, 3, 32, 694, 1, 0, 0, 0, 6, 150, 6578, 1, 0, 0, 0, 4, 20, 1524, 72792, 1, 0, 0, 0, 0, 10, 270, 12600, 920904, 1, 0, 0, 0, 0, 5, 40, 1764, 147328, 13109088, 1, 0, 0, 0, 0, 0, 15, 210, 12600, 1705536, 207360912, 1, 0, 0, 0, 0, 0, 6, 70, 2464, 146880, 23681520, 3608233056
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2022

Keywords

Examples

			Square array begins:
     1,    1,   1,  1,  1, 1, 1, ...
     1,    0,   0,  0,  0, 0, 0, ...
     3,    2,   0,  0,  0, 0, 0, ...
    14,    3,   3,  0,  0, 0, 0, ...
    88,   32,   6,  4,  0, 0, 0, ...
   694,  150,  20, 10,  5, 0, 0, ...
  6578, 1524, 270, 40, 15, 6, 0, ...

Links

Crossrefs

Columns k=0..3 give A007840, A052830, A351505, A351506.
Cf. A355610, A355665.

Programs

  • Mathematica
    T[n_, k_] := n! * Sum[j! * Abs[StirlingS1[n - k*j, j]]/(k!^j*(n - k*j)!), {j, 0, Floor[n/(k + 1)]}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 13 2022 *)
  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(k!^j*(n-k*j)!));

Formula

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

A355507 Expansion of e.g.f. (1 - x)^(-x^4/24).

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 15, 70, 420, 3024, 28350, 272250, 2875950, 33333300, 420840420, 5763671550, 84799915200, 1334007397800, 22343877115560, 396971840865600, 7456250728017000, 147612122975772000, 3071792315894841000, 67030983483724953000, 1530448652869851191400
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Crossrefs

Column k=4 of A355610.
Cf. A351493.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^4/24)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^4/24*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=5, i, j/(j-4)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\5, abs(stirling(n-4*k, k, 1))/(24^k*(n-4*k)!));

Formula

a(0) = 1; a(n) = (n-1)!/24 * Sum_{k=5..n} k/(k-4) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/5)} |Stirling1(n-4*k,k)|/(24^k * (n-4*k)!).
a(n) ~ n! / (Gamma(1/24) * n^(23/24)). - Vaclav Kotesovec, Jul 21 2022

A355650 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/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, 3, 16, 52, 1, 0, 0, 0, 6, 65, 203, 1, 0, 0, 0, 4, 10, 336, 877, 1, 0, 0, 0, 0, 10, 105, 1897, 4140, 1, 0, 0, 0, 0, 5, 20, 651, 11824, 21147, 1, 0, 0, 0, 0, 0, 15, 35, 2968, 80145, 115975, 1, 0, 0, 0, 0, 0, 6, 35, 616, 18936, 586000, 678570
Offset: 0

Views

Author

Seiichi Manyama, Jul 12 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, ...
    5,   3,   3,  0,  0, 0, 0, ...
   15,  16,   6,  4,  0, 0, 0, ...
   52,  65,  10, 10,  5, 0, 0, ...
  203, 336, 105, 20, 15, 6, 0, ...

Crossrefs

Columns k=0..3 give A000110, A052506, A354000, A354001.
Cf. A145460, A292892, A355610.

Programs

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

Formula

T(0,k) = 1 and T(n,k) = ((n-1)!/k!) * 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))} Stirling2(n-k*j,j)/(k!^j * (n-k*j)!).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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