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.

A351761 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, 4, 0, 1, 3, 12, 21, 0, 1, 4, 24, 102, 148, 0, 1, 5, 40, 279, 1160, 1305, 0, 1, 6, 60, 588, 4332, 16490, 13806, 0, 1, 7, 84, 1065, 11536, 84075, 281292, 170401, 0, 1, 8, 112, 1746, 25220, 282900, 1958058, 5598110, 2403640, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2022

Keywords

Examples

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  0,    1,     2,     3,      4,      5, ...
  0,    4,    12,    24,     40,     60, ...
  0,   21,   102,   279,    588,   1065, ...
  0,  148,  1160,  4332,  11536,  25220, ...
  0, 1305, 16490, 84075, 282900, 746525, ...

Crossrefs

Columns k=0..3 give A000007, A006153, A351762, A351763.
Main diagonal gives A351765.
Cf. A292860, A351776.

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.

A351778 Expansion of e.g.f. 1/(1 + 3*x*exp(x)).

Original entry on oeis.org

1, -3, 12, -63, 420, -3435, 33462, -379155, 4903896, -71318259, 1152202290, -20474486043, 396890715636, -8334602179995, 188486823883134, -4567087352339235, 118039115079323952, -3241465018561379427, 94249758656850366186, -2892678859033260044043
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Crossrefs

Column k=3 of A351776.
Cf. A351763.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1+3x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 21 2024 *)
  • 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

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.

A351779 a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * (n-k)^k/k!.

Original entry on oeis.org

1, -1, 4, -63, 2288, -138525, 12381084, -1528482823, 249005711296, -51739455340953, 13353206066063900, -4190486732316600771, 1571373340568392914288, -693899460077821703051125, 356404409990391961980227068, -210670220153918100996704166975
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Crossrefs

Main diagonal of A351776.
Cf. A351765, A351780.

Programs

  • 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)).

A351791 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, 0, 6, 1, 1, -2, -3, 24, 1, 1, -4, -6, -4, 120, 1, 1, -6, -3, 40, 25, 720, 1, 1, -8, 6, 132, 120, 114, 5040, 1, 1, -10, 21, 248, -375, -1872, -287, 40320, 1, 1, -12, 42, 364, -2120, -8298, -3920, -4152, 362880, 1, 1, -14, 69, 456, -5655, -12144, 86121, 155776, -1647, 3628800
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,  0,  -2,   -4,    -6,    -8, ...
    6, -3,  -6,   -3,     6,    21, ...
   24, -4,  40,  132,   248,   364, ...
  120, 25, 120, -375, -2120, -5655, ...

Crossrefs

Columns k=0..4 give A000142, (-1)^n * A302397(n), A336959, A351792, A351793.
Main diagonal gives (-1)^n * A302398(n).
Cf. A351776, A351790.

Programs

  • Mathematica
    T[n_, k_] := n!*(1 + Sum[(-k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 10}, {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.

A351777 Expansion of e.g.f. 1/(1 + 2*x*exp(x)).

Original entry on oeis.org

1, -2, 4, -6, -8, 150, -972, 3682, 6256, -289746, 3300460, -21071622, -27876312, 3156947014, -53217341660, 494232431250, 175171749088, -113735274256290, 2613309376750812, -32653995355358678, 36013529538641560, 10227377502146048118, -305630239215263764076
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Crossrefs

Column k=2 of A351776.
Cf. A351762.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1+2x Exp[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 06 2024 *)
  • PARI
    my(N=40, 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.
Showing 1-5 of 5 results.

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