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.

A351762 Expansion of e.g.f. 1/(1 - 2*x*exp(x)).

Original entry on oeis.org

1, 2, 12, 102, 1160, 16490, 281292, 5598110, 127326096, 3257961426, 92625793940, 2896747456262, 98827517418456, 3652643136982970, 145385563800940764, 6200097935648462190, 282035994269804870432, 13631368700936950044578, 697586352315912913754916
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2022

Keywords

Crossrefs

Column k=2 of A351761.
Cf. A000629, A351777.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*x*exp(x))))
  • PARI
    a(n) = n!*sum(k=0, n, 2^(n-k)*(n-k)^k/k!);
  • PARI
    a(n) = if(n==0, 1, 2*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

Formula

a(n) = n! * Sum_{k=0..n} 2^(n-k) * (n-k)^k/k!.
a(0) = 1 and a(n) = 2 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) ~ n! / ((1 + LambertW(1/2)) * LambertW(1/2)^n). - Vaclav Kotesovec, Feb 19 2022

A351703 Square array T(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 - x^k * exp(x) / k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 4, 1, 0, 1, 21, 1, 0, 0, 3, 148, 1, 0, 0, 1, 12, 1305, 1, 0, 0, 0, 4, 70, 13806, 1, 0, 0, 0, 1, 10, 465, 170401, 1, 0, 0, 0, 0, 5, 40, 3591, 2403640, 1, 0, 0, 0, 0, 1, 15, 315, 31948, 38143377, 1, 0, 0, 0, 0, 0, 6, 35, 2296, 319068, 672552730
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2022

Keywords

Examples

			Square array begins:
      1,   1,  1,  1, 1, 1, ...
      1,   0,  0,  0, 0, 0, ...
      4,   1,  0,  0, 0, 0, ...
     21,   3,  1,  0, 0, 0, ...
    148,  12,  4,  1, 0, 0, ...
   1305,  70, 10,  5, 1, 0, ...
  13806, 465, 40, 15, 6, 1, ...

Crossrefs

Column k=1..5 gives A006153, A346888, A346889, A346890, A346893.
Cf. A143398, A351761.

Programs

  • PARI
    T(n, k) = if(n==0, 1, binomial(n, k)*sum(j=0, n-k, binomial(n-k, j)*T(j, k)));
  • PARI
    T(n, k) = n!*sum(j=0, n\k, j^(n-k*j)/(k!^j*(n-k*j)!)); \\ Seiichi Manyama, May 13 2022

Formula

T(0,k) = 1 and T(n,k) = binomial(n,k) * Sum_{j=0..n-k} binomial(n-k,j) * T(j,k) for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/k)} j^(n-k*j)/(k!^j * (n-k*j)!). - Seiichi Manyama, May 13 2022

A351776 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * (n-j)^j/j!.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,     1,      1,       1, ...
  0,  -1,  -2,    -3,     -4,      -5, ...
  0,   0,   4,    12,     24,      40, ...
  0,   3,  -6,   -63,   -204,    -465, ...
  0,  -4,  -8,   420,   2288,    7180, ...
  0, -25, 150, -3435, -32020, -138525, ...

Crossrefs

Columns k=0..3 give A000007, A302397, A351777, A351778.
Main diagonal gives A351779.
Cf. A292861, A351761.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 + k*x*exp(x)).
T(0,k) = 1 and T(n,k) = -k * n * Sum_{j=0..n-1} binomial(n-1,j) * T(j,k) for n > 0.

A351765 a(n) = n! * Sum_{k=0..n} n^(n-k) * (n-k)^k/k!.

Original entry on oeis.org

1, 1, 12, 279, 11536, 746525, 69768036, 8902181575, 1487939919936, 315597946293657, 82839437215344100, 26366747854082944451, 10006618140321691249296, 4464690010732922712332149, 2313871692128866349730705924, 1378552938661073773617331110975
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2022

Keywords

Crossrefs

Main diagonal of A351761.
Cf. A332408, A351768, A351779.

Programs

  • Mathematica
    Join[{1}, Table[n!*Sum[n^(n - k)*(n - k)^k/k!, {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Feb 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, n^(n-k)*(n-k)^k/k!);

Formula

a(n) = n! * [x^n] 1/(1 - n*x*exp(x)).
From Vaclav Kotesovec, Feb 19 2022: (Start)
a(n) ~ exp(1) * n! * n^n.
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n-1). (End)

A351790 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (k * (n-j))^j/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 6, 21, 24, 1, 1, 8, 42, 148, 120, 1, 1, 10, 69, 392, 1305, 720, 1, 1, 12, 102, 780, 4600, 13806, 5040, 1, 1, 14, 141, 1336, 11145, 64752, 170401, 40320, 1, 1, 16, 186, 2084, 22200, 191178, 1063216, 2403640, 362880
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,     1,     1,     1, ...
    2,    4,    6,     8,    10,    12, ...
    6,   21,   42,    69,   102,   141, ...
   24,  148,  392,   780,  1336,  2084, ...
  120, 1305, 4600, 11145, 22200, 39145, ...

Crossrefs

Columns k=0..4 give A000142, A006153, A336950, A336951, A336952.
Main diagonal gives A235328.
Cf. A351761, A351791.

Programs

  • Mathematica
    T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));

Formula

E.g.f. of column k: 1/(1 - x*exp(k*x)).
T(0,k) = 1 and T(n,k) = n * Sum_{j=0..n-1} k^(n-1-j) * binomial(n-1,j) * T(j,k) for n > 0.

A351763 Expansion of e.g.f. 1/(1 - 3*x*exp(x)).

Original entry on oeis.org

1, 3, 24, 279, 4332, 84075, 1958058, 53202387, 1652070696, 57713665779, 2240196853710, 95650311987483, 4455281606078988, 224815388384744859, 12216916158370619010, 711312392929267383075, 44176151714082889756368, 2915038701200389804440675
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2022

Keywords

Crossrefs

Column k=3 of A351761.
Cf. A351778.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*exp(x))))
  • PARI
    a(n) = n!*sum(k=0, n, 3^(n-k)*(n-k)^k/k!);
  • PARI
    a(n) = if(n==0, 1, 3*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

Formula

E.g.f.: 1/(1 - 3*x*exp(x)).
a(n) = n! * Sum_{k=0..n} 3^(n-k) * (n-k)^k/k!.
a(0) = 1 and a(n) = 3 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) ~ n! / ((1 + LambertW(1/3)) * LambertW(1/3)^n). - Vaclav Kotesovec, Feb 19 2022
Showing 1-6 of 6 results.

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