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.

A052862 Expansion of e.g.f. log(-1/(-2+exp(x)))*x.

Original entry on oeis.org

0, 0, 2, 6, 24, 130, 900, 7574, 74928, 851274, 10916700, 155919742, 2453941512, 42188446898, 786563892660, 15805750451430, 340522975054176, 7829628493247002, 191363568551328780, 4954089147107164238
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Comments

A simple grammar.
For n > 2, a(n) = 2 * n * A000670(n-2). - Gerald McGarvey, Nov 01 2007 [corrected by Seiichi Manyama, May 26 2022]

Crossrefs

Programs

  • Maple
    spec := [S,{B=Cycle(C),C=Set(Z,1 <= card),S=Prod(Z,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*((i+r)^(n-r)/(i!*(k-i-r)!)), {i, 0, k-r}], {k, r, n}];
    Fubini[0, 1] = 1;
    a[n_] := If[n == 2, 2, 2 n * Fubini[n-2, 1]];
    Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 11 2022 *)
  • PARI
    my(x='x+O('x^25)); concat([0,0],Vec(serlaplace(log(-1/(-2+exp(x)))*x))) \\ Joerg Arndt, Oct 11 2022

Formula

a(n) ~ (n-1)! / log(2)^(n-1). - Vaclav Kotesovec, Aug 04 2014

A354412 Expansion of e.g.f. 1/(2 - exp(x))^(x/2).

Original entry on oeis.org

1, 0, 1, 3, 15, 95, 735, 6727, 71169, 854919, 11497845, 171179261, 2795081751, 49668211177, 954226247247, 19709181213555, 435524370171393, 10252531220906051, 256148413939459917, 6769302493147288885, 188664988853982963735, 5530544750788380455433
Offset: 0

Views

Author

Seiichi Manyama, May 25 2022

Keywords

Crossrefs

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(2-Exp[x])^(x/2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 12 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(2-exp(x))^(x/2)))
    
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, (k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])/2); v;

Formula

a(0) = 1; a(n) = (1/2) * Sum_{k=1..n} A052862(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ n! / (Gamma(log(2)/2) * 2^(log(2)/2) * n^(1 - log(2)/2) * log(2)^(n + log(2)/2)). - Vaclav Kotesovec, Jun 08 2022

A367486 Expansion of e.g.f. 1/(3 - 2*exp(x))^x.

Original entry on oeis.org

1, 0, 4, 18, 168, 1830, 24540, 388122, 7084560, 146650446, 3395460900, 86962122786, 2441210321880, 74542218945558, 2459830123779756, 87236196407090730, 3308881779086345760, 133667058288336876894, 5729380391745420070068
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, 2^k*(k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A367489(k) * binomial(n-1,k-1) * a(n-k).

A367488 Expansion of e.g.f. 1/(4 - 3*exp(x))^x.

Original entry on oeis.org

1, 0, 6, 36, 444, 6540, 119520, 2593164, 65233392, 1867289868, 59939612040, 2132540249532, 83293357351248, 3543242182036284, 163062595422642552, 8071964230348189260, 427682380939864204224, 24149065480351703398572, 1447640087400503974386504
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, j*sum(k=1, j-1, 3^k*(k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A367490(k) * binomial(n-1,k-1) * a(n-k).

A354421 Expansion of e.g.f. (2 - exp(x))^x.

Original entry on oeis.org

1, 0, -2, -6, -12, -10, 60, 406, 672, -18666, -400740, -6617842, -108686952, -1883464466, -34930602252, -693981413610, -14732243810016, -333084114060442, -7994768036250132, -203102355108133154, -5445884954606704920, -153726156157794541986
Offset: 0

Views

Author

Seiichi Manyama, May 26 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((2-exp(x))^x))
    
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, j*sum(k=1, j-1, (k-1)!*stirling(j-1, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} A052862(k) * binomial(n-1,k-1) * a(n-k).
a(n) ~ -n! / (Gamma(1 - log(2)) * 2^(-log(2)) * n^(log(2) + 1) * log(2)^(n - log(2) - 1)). - Vaclav Kotesovec, Jun 08 2022
Showing 1-5 of 5 results.