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 1-8 of 8 results.

A377530 Expansion of e.g.f. 1/(1 - x * exp(x))^3.

Original entry on oeis.org

1, 3, 18, 141, 1380, 16095, 217458, 3335745, 57225528, 1085066523, 22526087070, 508042140573, 12367076890644, 323130848000727, 9018976230237834, 267789942962863065, 8427492557547704688, 280194087519310655667, 9813332205452943323190, 361109786425470021564021
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A006153, A377529.
Cf. A377532, A377534.
Cf. A377504.

Programs

  • Mathematica
    nmax=19; CoefficientList[Series[1/(1 - x * Exp[x])^3,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 05 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+2,2)/(n-k)!.
a(n) ~ n! * n^2 / (2 * (1+LambertW(1))^3 * LambertW(1)^n). - Vaclav Kotesovec, Oct 31 2024

A379933 Expansion of e.g.f. 1/( exp(-x) - x )^2.

Original entry on oeis.org

1, 4, 22, 158, 1408, 15002, 186100, 2634998, 41937136, 741170834, 14402727484, 305225470046, 7005711916840, 173134991854970, 4583675648417044, 129424786945875398, 3882446011526729440, 123304773913531035170, 4133369745467043807340, 145840627118145774415214
Offset: 0

Views

Author

Seiichi Manyama, Jan 06 2025

Keywords

Links

Crossrefs

Cf. A072597, A379942, A379943.
Cf. A379934, A379936.
Cf. A358738, A377529.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A072597.
a(n) = n! * Sum_{k=0..n} (k+1) * (k+2)^(n-k)/(n-k)!.

A379993 Expansion of e.g.f. 1/(1 - x * exp(x))^4.

Original entry on oeis.org

1, 4, 28, 252, 2776, 35940, 533304, 8908228, 165247072, 3368072196, 74782987240, 1796037420804, 46379441090448, 1281203788073092, 37694510810334616, 1176606639075726660, 38833052393329645504, 1351066066253778043908, 49417629820950190273992
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A379943, A379994, A379995, A379996.
Cf. A006153, A377529, A377530.

Programs

  • Mathematica
    nmax=18;CoefficientList[Series[1/(1 - x * Exp[x])^4,{x,0,nmax}],x]Range[0,nmax]! (* Stefano Spezia, Feb 05 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(k+3, 3)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(k+3,3)/(n-k)!.
a(n) == 0 (mod 4) for n>0.

A377527 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x)^2)^2.

Original entry on oeis.org

1, 2, 26, 618, 22256, 1081770, 66401532, 4931389358, 430108545680, 43104305664594, 4881518010253460, 616559703960596022, 85935621525038617752, 13102417265843584412474, 2169337115977056447577820, 387609934848899388554651550, 74340899731294447790784890912
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A377526, A377528.
Cf. A377503, A377529.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+1, k)/((2*k+1)*(n-k)!));

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377526.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+1,k)/( (2*k+1)*(n-k)! ).

A380841 Array read by ascending antidiagonals: A(n,k) = n! * [x^n] 1/(1 - x*exp(x))^k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 21, 10, 3, 1, 0, 148, 66, 18, 4, 1, 0, 1305, 560, 141, 28, 5, 1, 0, 13806, 5770, 1380, 252, 40, 6, 1, 0, 170401, 69852, 16095, 2776, 405, 54, 7, 1, 0, 2403640, 970886, 217458, 35940, 4940, 606, 70, 8, 1, 0, 38143377, 15228880, 3335745, 533304, 70045, 8088, 861, 88, 9, 1
Offset: 0

Views

Author

Stefano Spezia, Feb 05 2025

Keywords

Examples

			Array begins as:
  1,    1,    1,     1,     1,     1,      1, ...
  0,    1,    2,     3,     4,     5,      6, ...
  0,    4,   10,    18,    28,    40,     54, ...
  0,   21,   66,   141,   252,   405,    606, ...
  0,  148,  560,  1380,  2776,  4940,   8088, ...
  0, 1305, 5770, 16095, 35940, 70045, 124350, ...
  ...

Crossrefs

Cf. A380843 (antidiagonal sums).
Columns k=0..4 give A000007, A006153, A377529, A377530, A379993.
Rows n=0..2 give A000012, A001477, A028552.
Main diagonal gives A380842.
A(n,n+1) gives A213643(n+1).

Programs

  • Mathematica
    A[n_,k_]:=n!SeriesCoefficient[1/(1-x*Exp[x])^k,{x,0,n}]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten

Formula

A(n,k) = n! * Sum_{j=0..n} j^(n-j) * binomial(j+k-1,j)/(n-j)!. - Seiichi Manyama, Feb 06 2025

A379990 Expansion of e.g.f. exp(-2*x)/(exp(-x) - x)^3.

Original entry on oeis.org

1, 4, 25, 205, 2065, 24601, 337837, 5249581, 91006657, 1740663937, 36402220141, 826159146253, 20220201899377, 530828186303377, 14878044338021677, 443397290411503021, 14000282854007503105, 466866129420834410881, 16395362179348570608205, 604794784980600986425645
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A377530, A379942, A379991.
Cf. A092148, A377529.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)^(n-k)*binomial(k+2, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..n} (k+1)^(n-k) * binomial(k+2,2)/(n-k)!.
a(n) ~ n! * n^2 / (2 * (LambertW(1) + 1)^3 * LambertW(1)^(n+1)). - Vaclav Kotesovec, Jan 08 2025

A379992 Expansion of e.g.f. exp(-3*x)/(exp(-x) - x)^2.

Original entry on oeis.org

1, 1, 7, 41, 349, 3539, 42451, 585605, 9130297, 158692679, 3041499871, 63712004729, 1447946191957, 35479218963083, 932326476195115, 26153289728300909, 779995883104560241, 24644267406802467215, 822278654588440803511, 28891372907012629446881
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A358738, A377529, A379933, A379997.
Cf. A368266, A377530, A379994.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[-3x]/(Exp[-x]-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 14 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)*(k-1)^(n-k)/(n-k)!);

Formula

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

A379997 Expansion of e.g.f. 1/(exp(x) - x*exp(2*x))^2.

Original entry on oeis.org

1, 0, 6, 22, 224, 2138, 25732, 351846, 5458224, 94441042, 1803255404, 37652268014, 853321021192, 20858236815258, 546941712302052, 15313467390967222, 455933682027961184, 14383416438784605602, 479254037890010238172, 16817855455956128823486, 619953003446894086537656
Offset: 0

Views

Author

Seiichi Manyama, Jan 07 2025

Keywords

Crossrefs

Cf. A358738, A377529, A379933, A379992.
Cf. A092148.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (k+1)*(k-2)^(n-k)/(n-k)!);

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A092148.
a(n) = n! * Sum_{k=0..n} (k+1) * (k-2)^(n-k)/(n-k)!.
Showing 1-8 of 8 results.

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