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.

A377529 Expansion of e.g.f. 1/(1 - x * exp(x))^2.

Original entry on oeis.org

1, 2, 10, 66, 560, 5770, 69852, 970886, 15228880, 266006610, 5119447700, 107617719022, 2453167135608, 60268223308826, 1587381621990556, 44619277892537910, 1333135910963656352, 42189279001183102882, 1409741875877923927332, 49597905017847180008126
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A006153, A377530.
Cf. A377503, A377527.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1-x Exp[x])^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 04 2025 *)
  • PARI
    a(n) = n!*sum(k=0, n, (k+1)*k^(n-k)/(n-k)!);

Formula

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

A377742 E.g.f. satisfies A(x) = exp(x) / (1 - x * A(x))^2.

Original entry on oeis.org

1, 3, 23, 331, 7133, 205901, 7470475, 326932299, 16768124217, 986753701657, 65548017270791, 4852285640543639, 396133183892522389, 35359325061987638661, 3426053898460864501251, 358128187005971803014211, 40172982580368589391407217, 4813677071886578522596221233
Offset: 0

Views

Author

Seiichi Manyama, Nov 05 2024

Keywords

Crossrefs

Cf. A194471, A377743, A377744.
Cf. A001339, A377745.
Cf. A377503.

Programs

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

Formula

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

A377504 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^3.

Original entry on oeis.org

1, 3, 36, 735, 21972, 871995, 43308378, 2588123811, 180990517032, 14507325973395, 1311719669172750, 132102208441613883, 14666354372331521676, 1779817542971018697003, 234399632982398657764578, 33297612755940733707395955, 5075234637265322738651060688, 826215756199826873368252279971
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377503, A377528.
Cf. A006632, A364987.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(4*k+2, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A364987.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(4*k+2,k)/( (k+1)*(n-k)! ).

A377528 E.g.f. satisfies A(x) = 1/(1 - x * exp(x) * A(x))^4.

Original entry on oeis.org

1, 4, 60, 1548, 58456, 2930020, 183763704, 13866109012, 1224251041248, 123885272536452, 14140672597851880, 1797709847594145364, 251941291752251706576, 38593132701417704324356, 6415647343472197357272984, 1150373241484390263973203540, 221318733487356013660505462464
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2024

Keywords

Crossrefs

Cf. A295238, A377503, A377504.
Cf. A377526, A377527.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, k^(n-k)*binomial(5*k+3, k)/((k+1)*(n-k)!));

Formula

E.g.f.: B(x)^4, where B(x) is the e.g.f. of A377526.
a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(5*k+3,k)/( (k+1)*(n-k)! ).

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)! ).
Showing 1-5 of 5 results.

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