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 1-4 of 4 results.

A383381 Expansion of e.g.f. exp(-2*x) / (1-x)^5.

Original entry on oeis.org

1, 3, 14, 82, 576, 4688, 43264, 445632, 5062016, 62812288, 844863744, 12239474432, 189939644416, 3142842052608, 55223903596544, 1026805938614272, 20139224002953216, 415503046091767808, 8994794537935765504, 203848794955954716672, 4826475681472562855936, 119162892472107134353408
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Crossrefs

Cf. A001909, A383382, A383383, A383384.
Cf. A000023, A052124, A087981, A383380.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-2*x)/(1-x)^5))

Formula

a(n) = n! * Sum_{k=0..n} (-2)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = 1, a(1) = 3; a(n) = (n+2)*a(n-1) + 2*(n-1)*a(n-2).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+2)). - Vaclav Kotesovec, Apr 25 2025

A383382 Expansion of e.g.f. exp(-3*x) / (1-x)^5.

Original entry on oeis.org

1, 2, 9, 48, 321, 2502, 22329, 223668, 2481921, 30187242, 399071529, 5694475608, 87197543361, 1425766728942, 24787205125209, 456477484618908, 8875541469155841, 181670665706512722, 3904395263350689609, 87898121215165479168, 2068411075529464370241, 50778930934558144895382
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Crossrefs

Cf. A001909, A383381, A383383, A383384.
Cf. A010843, A137775, A383378.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-3*x)/(1-x)^5))

Formula

a(n) = n! * Sum_{k=0..n} (-3)^(n-k) * binomial(k+4,4)/(n-k)!.
a(0) = 1, a(1) = 2; a(n) = (n+1)*a(n-1) + 3*(n-1)*a(n-2).
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+3)). - Vaclav Kotesovec, Apr 25 2025

A383384 Expansion of e.g.f. exp(-5*x) / (1-x)^5.

Original entry on oeis.org

1, 0, 5, 10, 105, 620, 5725, 52950, 571025, 6686200, 85871925, 1193029250, 17846277625, 285737086500, 4874590170125, 88245858436750, 1689282139310625, 34088182903910000, 723088091207873125, 16083522103093616250, 374280288623526655625, 9093957982779894737500
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Crossrefs

Column k=5 of A295181.
Cf. A001909, A383381, A383382, A383383.
Cf. A000166.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[-5x]/(1-x)^5,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 04 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-5*x)/(1-x)^5))

Formula

a(n) = n! * Sum_{k=0..n} (-5)^(n-k) * binomial(k+4,4)/(n-k)!.
a(n) = (n-1) * (a(n-1) + 5*a(n-2)) for n > 1.
E.g.f.: B(x)^5, where B(x) is the e.g.f. of A000166.
a(n) ~ sqrt(2*Pi) * n^(n + 9/2) / (24*exp(n+5)). - Vaclav Kotesovec, Apr 25 2025

A383341 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where A(n,k) = n! * Sum_{j=0..n} (-k)^(n-j) * binomial(j+k,j)/(n-j)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 4, 11, 24, 1, 1, 5, 16, 53, 120, 1, 1, 6, 21, 88, 309, 720, 1, 1, 7, 26, 129, 568, 2119, 5040, 1, 1, 8, 31, 176, 897, 4288, 16687, 40320, 1, 1, 9, 36, 229, 1296, 7317, 36832, 148329, 362880, 1, 1, 10, 41, 288, 1765, 11296, 67365, 354688, 1468457, 3628800
Offset: 0

Views

Author

Seiichi Manyama, Apr 24 2025

Keywords

Examples

			Square array begins:
    1,    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,    1,     1,     1,     1, ...
    2,    3,    4,    5,     6,     7,     8, ...
    6,   11,   16,   21,    26,    31,    36, ...
   24,   53,   88,  129,   176,   229,   288, ...
  120,  309,  568,  897,  1296,  1765,  2304, ...
  720, 2119, 4288, 7317, 11296, 16315, 22464, ...

Crossrefs

Columns k=0..4 give A000142, A000255, A052124, A383378, A383383.
Main diagonal gives A383379.
Cf. A295181.

Programs

  • PARI
    a(n,k) = n!*sum(j=0, n, (-k)^(n-j)*binomial(j+k, j)/(n-j)!);

Formula

E.g.f. of column k: exp(-k*x) / (1-x)^(k+1).
A(0,k) = A(1,k) = 1; A(n,k) = n*A(n-1,k) + k*(n-1)*A(n-2,k).
Showing 1-4 of 4 results.

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

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