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 1-7 of 7 results.

A382036 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * C(x)^2) ), where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.

Original entry on oeis.org

1, 1, 7, 94, 1901, 51696, 1771267, 73317616, 3560476761, 198531343360, 12502959204671, 877829600807424, 67991178144166213, 5759309535250776064, 529665762441463234875, 52560256640090731902976, 5597859153748148214250673, 636915477940535101583130624, 77102760978489789146276986231
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2025

Keywords

Crossrefs

Cf. A052873, A382037, A382038.
Cf. A000108, A377829, A382032.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp(x*A(x) * C(x*A(x))^2).
a(n) = (n-1)! * Sum_{k=0..n-1} (n+1)^(n-k-1) * binomial(2*n,k)/(n-k-1)! for n > 0.
E.g.f.: exp( Series_Reversion( x*exp(-x)/(1+x)^2 ) ).
a(n) ~ 2^(n - 1/2) * n^(n-1) / ((sqrt(2) - 1)^(n - 1/2) * exp((sqrt(2) - 1)*(sqrt(2)*n - 1))). - Vaclav Kotesovec, Mar 15 2025

A380781 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * (1 + x)^2) / (1 + x)^2 ).

Original entry on oeis.org

1, 3, 29, 514, 13473, 470616, 20607781, 1086800352, 67105960641, 4750972007680, 379512594172941, 33771911612182272, 3313441417839023521, 355371388642280715264, 41365962922892138767125, 5193995331631149377867776, 699785874809076112607739009, 100701968551637581411176480768
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Links

Crossrefs

Cf. A377829, A380778, A380779, A380780.
Cf. A365031, A380762.
Cf. A380666.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + x*A(x))^2 ) * (1 + x*A(x))^2.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n+2*k+2,n-k)/k!.

A377830 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x)/(1 + x)^3 ).

Original entry on oeis.org

1, 4, 45, 886, 25397, 963216, 45615553, 2595412240, 172624541769, 13150155923200, 1129371806449301, 107987110491257856, 11379014255782146685, 1310277285293012678656, 163703077517048727256425, 22057132253723442887059456, 3188342874266180285119069457, 492178313447920665621400780800
Offset: 0

Views

Author

Seiichi Manyama, Nov 09 2024

Keywords

Links

Crossrefs

Cf. A088690, A377829.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^3 * exp(x * A(x)).
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(3*n+3,n-k)/k!.

A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

Original entry on oeis.org

1, 4, 46, 932, 27568, 1080432, 52916176, 3115326496, 214470890496, 16914853191680, 1504252282653184, 148956086481767424, 16256865070022066176, 1938988214539948730368, 250943399365390735104000, 35026523834624205803491328, 5245178283068781060488298496, 838841884254236846183525646336
Offset: 0

Views

Author

Seiichi Manyama, Feb 06 2025

Keywords

Links

Crossrefs

Cf. A088690, A380647, A380648.
Cf. A065866, A377829.
Cf. A097629, A380828.
Cf. A377892.

Programs

  • Mathematica
    nmax=18; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-2*x]/(1 + x)^2 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)
  • PARI
    a(n) = 2*n!*sum(k=0, n, (2*n+2)^(k-1)*binomial(2*n+2, n-k)/k!);

Formula

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(2 * x * A(x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377892.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(2*n+2,n-k)/k!.

A380778 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x / (1 + x)^2) / (1 + x)^2 ).

Original entry on oeis.org

1, 3, 21, 238, 3777, 77616, 1966381, 59379888, 2085295617, 83580555520, 3767468068581, 188731359078912, 10405256927541889, 626236791181897728, 40860738460515664125, 2873352871221375440896, 216652727562188159522049, 17437704874236857627246592, 1492289181734461545084103477
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Links

Crossrefs

Cf. A377829, A380779, A380780, A380781.
Cf. A380674.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x * A(x) / (1 + x*A(x))^2 ) * (1 + x*A(x))^2.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n-2*k+2,n-k)/k!.

A380779 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x / (1 + x)) / (1 + x)^2 ).

Original entry on oeis.org

1, 3, 23, 298, 5529, 134496, 4062631, 146903184, 6193969137, 298577002240, 16204658051031, 978156957629952, 65017249611283657, 4719532271850590208, 371519503997940966375, 31526820740816885549056, 2869134152226896957509089, 278763390556764407051452416
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Links

Crossrefs

Cf. A377829, A380778, A380780, A380781.
Cf. A380675.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x * A(x) / (1 + x*A(x)) ) * (1 + x*A(x))^2.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n-k+2,n-k)/k!.

A380780 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * (1 + x)) / (1 + x)^2 ).

Original entry on oeis.org

1, 3, 27, 436, 10353, 326856, 12920731, 614694816, 34223383809, 2184028353280, 157223422977531, 12606338448248832, 1114292924502666673, 107657947282494206976, 11287975339133863810875, 1276603658863119005618176, 154909721707963344338403969, 20076669149268201122957819904
Offset: 0

Views

Author

Seiichi Manyama, Feb 02 2025

Keywords

Links

Crossrefs

Cf. A377829, A380778, A380779, A380781.
Cf. A380665.

Programs

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

Formula

E.g.f. A(x) satisfies A(x) = exp( x * A(x) * (1 + x*A(x)) ) * (1 + x*A(x))^2.
a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*n+k+2,n-k)/k!.
Showing 1-7 of 7 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.