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.

Previous Showing 21-27 of 27 results.

A368266 Expansion of e.g.f. exp(-2*x) / (1 - x*exp(x)).

Original entry on oeis.org

1, -1, 4, 1, 60, 393, 4498, 54689, 773384, 12268369, 216328614, 4195769433, 88776964444, 2034936319817, 50232646818890, 1328570248040497, 37481046375146640, 1123486426007081505, 35657224567565828302, 1194561018775753556777, 42125545306641497600036
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A006153, A072597, A092148, A368265.

Programs

  • Mathematica
    CoefficientList[Series[Exp[-2*x]/ (1 - x*Exp[x]),{x,0,20}],x]Range[0,20]! (* Stefano Spezia, Feb 21 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n-k-2)^k/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n-k-2)^k / k!.

A375395 Expansion of e.g.f. 1 / (exp(-x^3/6) - x).

Original entry on oeis.org

1, 1, 2, 7, 32, 180, 1210, 9520, 85680, 867160, 9749600, 120582000, 1626994600, 23782158400, 374367193200, 6314037129400, 113591474796800, 2171267969270400, 43944509528920000, 938808209417478400, 21111813400597920000, 498498097342637392000
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Cf. A072597, A375394.

Programs

  • Maple
    A375395 := proc(n)
    n!*add(((n-3*k+1)/6)^k/k!,k=0..floor(n/3)) ;
end proc:
seq(A375395(n),n=0..60) ; # R. J. Mathar, Aug 23 2024
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x^3/6)-x)))
  • PARI
    a(n) = n!*sum(k=0, n\3, ((n-3*k+1)/6)^k/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} ((n-3*k+1)/6)^k/k!.

A377741 E.g.f. satisfies A(x) = exp(x) * (1 + x * A(x))^3.

Original entry on oeis.org

1, 4, 37, 583, 13225, 394681, 14659537, 652829857, 33937422001, 2018665692721, 135274646371561, 10087017309339433, 828563190097478425, 74348364577760978329, 7236649495742795579809, 759466703902106082652321, 85492204279344776678878945, 10275933748282019792253453025
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2024

Keywords

Crossrefs

Cf. A072597, A377740.
Cf. A364983, A377575.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k-1)*binomial(3*k+3, k)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k-1) * binomial(3*k+3,k)/(n-k)!.
a(n) = A364983(n+1)/(n+1).

A305133 E.g.f.: (1-x) / (exp(-x) - x).

Original entry on oeis.org

1, 1, 3, 16, 113, 996, 10537, 130054, 1834513, 29111896, 513307601, 9955832514, 210652214665, 4828548335092, 119193293536969, 3152465052989326, 88935973854834593, 2665836978234855984, 84608363388300429601, 2834484567764492239354, 99956558270008377397081, 3701159405682998540166796, 143571313108884280622221913, 5822409005523822986360056326
Offset: 0

Views

Author

Paul D. Hanna, Jun 15 2018

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 113*x^4/4! + 996*x^5/5! + 10537*x^6/6! + 130054*x^7/7! + 1834513*x^8/8! + 29111896*x^9/9! + ...
RELATED TABLE.
The table of coefficients of x^k in exp(n*x) * A(x) begins:
n=0: [1, (1), 3/2, 8/3, 113/24, 83/10, 10537/720, 65027/2520, ...];
n=1: [(1), 2, (3), 29/6, 25/3, 1757/120, 929/36, 45863/1008, ...];
n=2: [1, (3), 11/2, (9), 361/24, 1559/60, 729/16, 101107/1260, ...];
n=3: [1, 4, (9), 97/6, (82/3), 1863/40, 3637/45, 714319/5040, ...];
n=4: [1, 5, 27/2, (82/3), 1169/24, (251/3), 103801/720, 632897/2520, ...];
n=5: [1, 6, 19, 87/2, (251/3), 17821/120, (5147/20), 2250499/5040, ...];
n=6: [1, 7, 51/2, 197/3, 3305/24, (5147/20), 65633/144, (14293/18), ...];
n=7: [1, 8, 33, 569/6, 652/3, 51893/120, (14293/18), 7078303/5040, ...]; ...
in which terms along the diagonals (enclosed in parenthesis) are equal:
[x^n] exp((n+1)*x) * A(x) = [x^(n+1)] exp(n*x) * A(x) for n >= 0.

Crossrefs

Cf. A072597, A305990.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1-x)/(Exp[-x]-x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 15 2022 *)
  • PARI
    {a(n) = n!*polcoeff( (1-x) / (exp(-x +x*O(x^n)) - x), n)}
for(n=0,30,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies: [x^n] exp((n+1)*x) * A(x) = [x^(n+1)] exp(n*x) * A(x) for n >= 0.
a(n) ~ n! * (1 - LambertW(1)) / ((1 + LambertW(1)) * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jun 16 2018
a(n) = 1 + n * Sum_{k=1..n-1} binomial(n-1,k) * a(k). - Ilya Gutkovskiy, Aug 08 2020

A368237 Expansion of e.g.f. 1/(exp(-x) - 3*x).

Original entry on oeis.org

1, 4, 31, 361, 5605, 108781, 2533447, 68836279, 2137543177, 74673228457, 2898494302651, 123757822391083, 5764497138070381, 290878956151681405, 15806942065094830735, 920336494043393536591, 57157621592164505969425, 3771643127452655490322513
Offset: 0

Views

Author

Seiichi Manyama, Dec 18 2023

Keywords

Crossrefs

Cf. A072597, A368236.
Cf. A336948.

Programs

  • Mathematica
    a[n_] := n! Sum[3^(n - k) (n - k + 1)^k / k!, {k, 0, n}];Table[a[n],{n,0,17}] (* or *) a[0] = 1;a[n_] := 3n a[n - 1] + Sum[(-1)^(k - 1) Binomial[n, k] a[n - k], {k, 1, n}];Table[a[n],{n,0,17}] (* James C. McMahon, Dec 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, 3^(n-k)*(n-k+1)^k/k!);
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(exp(-x) - 3*x))) \\ Michel Marcus, Dec 18 2023

Formula

a(0) = 1; a(n) = 3*n*a(n-1) + Sum_{k=1..n} (-1)^(k-1) * binomial(n,k) * a(n-k).
a(n) = n! * Sum_{k=0..n} 3^(n-k) * (n-k+1)^k / k!.

A375610 Expansion of e.g.f. 1 / (exp(-x) - x^3).

Original entry on oeis.org

1, 1, 1, 7, 49, 241, 1681, 18481, 192193, 2028097, 26854561, 400419361, 6074016961, 100260498625, 1847840462833, 36061045391281, 738757221740161, 16244778936351361, 380460397886975809, 9341152506044172865, 241084169507148900481, 6559259107807215358081
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Cf. A072597, A352303.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(exp(-x)-x^3)))
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+1)^(n-3*k)/(n-3*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(n-3*k)/(n-3*k)!.
a(n) == 1 (mod 6).
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / ((1 + LambertW(1/3)) * 3^(n+4) * exp(n) * LambertW(1/3)^(n+3)). - Vaclav Kotesovec, Aug 21 2024

A341093 Triangular array read by rows. T(n,k) is the number of partial functions on [n] with index k, n=0 implies k=1, otherwise n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 2, 7, 2, 37, 21, 6, 261, 232, 108, 24, 2301, 2935, 1760, 660, 120, 24343, 42396, 30630, 14880, 4680, 720, 300455, 692055, 586572, 335790, 139440, 37800, 5040, 4238153, 12631200, 12387592, 8008896, 3959760, 1438080, 342720, 40320
Offset: 0

Views

Author

Geoffrey Critzer, Feb 13 2022

Keywords

Comments

For every partial function f, there are smallest positive integers k,m such that f^k = f^(k+m). The integer k is the index of f.

Examples

			Array begins
      1;
      2;
      7,     2;
     37,    21,     6;
    261,   232,   108,    24;
   2301,  2935,  1760,   660,  120;
  24343, 42396, 30630, 14880, 4680, 720;
  ...

Crossrefs

Cf. A072597 (column k=1), A000169(n+1) (row sums).

Programs

  • Mathematica
    nn = 8; np = Exp[NestList[x Exp[#] &, x, nn]]; fp = Exp[Log[1/(1 - NestList[x Exp[#] &, x Exp[x], nn])]];Map[Select[#, # > 0 &] &,Prepend[Table[Range[0, nn]! CoefficientList[Series[(fp[[k + 1]] - fp[[k]])*(np[[k + 1]]) + (fp[[k + 1]])*(np[[k + 1]] - np[[k]]) - (fp[[k + 1]] - fp[[k]]) (np[[k + 1]] - np[[k]]), {x, 0, nn}], x], {k, 1, nn - 1}], Range[0, nn]! CoefficientList[Series[1/(1 - x Exp[x])*Exp[x], {x, 0, nn}], x]] // Transpose] // Grid
Previous Showing 21-27 of 27 results.

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