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.

A295182 a(n) = n! * [x^n] exp(-n*x)/(1 - x)^n.

Original entry on oeis.org

1, 0, 2, 6, 72, 620, 8640, 122346, 2156672, 41367672, 905126400, 21646532270, 570077595648, 16268377195044, 502096929431552, 16629319748711250, 588938142209310720, 22196966267762213744, 887352465220427317248, 37496112562144553167062, 1670071417348195942400000, 78195398849926292810318940
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 16 2017

Keywords

Comments

The n-th term of the n-fold exponential convolution of A000166 with themselves.

Links

Crossrefs

Main diagonal of A295181.
Cf. A000166, A000407, A001813, A087981, A137775, A295183, A383379.

Programs

  • Maple
    S:= series((exp(-x)/(1-x))^n,x,30):
seq(n!*coeff(S,x,n),n=0..29); # Robert Israel, Nov 16 2017
  • Mathematica
    Table[n! SeriesCoefficient[Exp[-n x]/(1 - x)^n, {x, 0, n}], {n, 0, 21}]

Formula

a(n) = A295181(n,n).
a(n) ~ phi^(3*n - 1/2) * n^n / (5^(1/4) * exp(n*(1 + 1/phi))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Nov 16 2017
a(n) = n! * Sum_{k=0..n} (-n)^(n-k) * binomial(n+k-1,k)/(n-k)!. - Seiichi Manyama, Apr 25 2025

A383344 Expansion of e.g.f. exp(-4*x) / (1-x)^4.

Original entry on oeis.org

1, 0, 4, 8, 72, 416, 3520, 31104, 316288, 3525632, 43117056, 572195840, 8191304704, 125761056768, 2060841582592, 35894401335296, 662066514984960, 12890305925218304, 264155723747688448, 5682905054074109952, 128051031032232411136, 3015653024970577018880
Offset: 0

Views

Author

Seiichi Manyama, Apr 23 2025

Keywords

Crossrefs

Column k=4 of A295181.
Cf. A000166, A087981, A137775.
Cf. A088991, A381504.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-4*x)/(1-x)^4))

Formula

a(n) = n! * Sum_{k=0..n} (-4)^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) = (n-1) * (a(n-1) + 4*a(n-2)) for n > 1.
E.g.f.: B(x)^4, where B(x) is the e.g.f. of A000166.
a(n) ~ sqrt(2*Pi) * n^(n + 7/2) / (6*exp(n+4)). - 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.

Content is available under The OEIS End-User License Agreement.