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.

A353344 Expansion of e.g.f. exp(-log(1 - x)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 210, 1710, 17304, 194712, 2402184, 32536080, 481094856, 7703580456, 132658888752, 2443228469136, 47904722262144, 995970495769920, 21879712141853760, 506301721998264000, 12306713585213260800, 313441368701926135680, 8345931596469584686080
Offset: 0

Views

Author

Seiichi Manyama, May 06 2022

Keywords

Links

Crossrefs

Column k=3 of A357882.
Cf. A000142, A353358, A353404.
Cf. A347002, A353118, A353664.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-log(1-x)^2)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i-1, j-1)*abs(stirling(j, 3, 1))*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*abs(stirling(n, 3*k, 1))/k!);

Formula

E.g.f.: (1 - x)^(-(log(1 - x))^2).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,3)| * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * |Stirling1(n,3*k)|/k!.

A353774 Expansion of e.g.f. 1/(1 - (exp(x) - 1)^3).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 1260, 16926, 197316, 2286150, 32821020, 548528046, 9515702196, 174531124950, 3521913283980, 76969474578366, 1777400236160676, 43405229295464550, 1126972561394470140, 30949983774936839886, 893095888222540548756, 27035433957000465352950
Offset: 0

Views

Author

Seiichi Manyama, May 07 2022

Keywords

Links

Crossrefs

Cf. A000670, A052841, A353775.
Cf. A143815, A346894, A353118, A353664.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-(exp(x)-1)^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*k)!*x^(3*k)/prod(j=1, 3*k, 1-j*x)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=6*sum(j=1, i, binomial(i, j)*stirling(j, 3, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*stirling(n, 3*k, 2));

Formula

G.f.: Sum_{k>=0} (3*k)! * x^(3*k)/Product_{j=1..3*k} (1 - j * x).
a(0) = 1; a(n) = 6 * Sum_{k=1..n} binomial(n,k) * Stirling2(k,3) * a(n-k).
a(n) = Sum_{k=0..floor(n/3)} (3*k)! * Stirling2(n,3*k).
a(n) ~ n! / (6 * log(2)^(n+1)). - Vaclav Kotesovec, May 08 2022

A353665 Expansion of e.g.f. exp((exp(x) - 1)^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 240, 1560, 8400, 60984, 912240, 15938520, 242998800, 3300493944, 44583979440, 690641504280, 12868117189200, 264164524958904, 5481631005177840, 112822632387018840, 2367468210865875600, 52624238539033647864, 1258531092544541563440
Offset: 0

Views

Author

Seiichi Manyama, May 07 2022

Keywords

Crossrefs

Cf. A000110, A052859, A353664.
Cf. A327505, A353358, A353775.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^4)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (4*k)!*x^(4*k)/(k!*prod(j=1, 4*k, 1-j*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=24*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 4, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\4, (4*k)!*stirling(n, 4*k, 2)/k!);

Formula

E.g.f.: exp((exp(x) - 1)^4).
G.f.: Sum_{k>=0} (4*k)! * x^(4*k)/(k! * Product_{j=1..4*k} (1 - j * x)).
a(0) = 1; a(n) = 24 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,4) * a(n-k).
a(n) = Sum_{k=0..floor(n/4)} (4*k)! * Stirling2(n,4*k)/k!.

A357025 E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^3.

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 3060, 62286, 867636, 15591750, 419764500, 10834588446, 277719263316, 8580282719190, 297021183388020, 10459810717672686, 393932179466738676, 16351788886638987750, 717798906181149294420, 32905220431196072057406
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2022

Keywords

Crossrefs

Cf. A030019, A357024.
Cf. A353664, A357010, A357032.

Programs

  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*(n+1)^(k-1)*stirling(n, 3*k, 2)/k!);

Formula

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (n+1)^(k-1) * Stirling2(n,3*k)/k!.

A357869 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} (k*j)!* Stirling2(n,k*j)/j!.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 0, 2, 5, 0, 1, 0, 0, 6, 15, 0, 1, 0, 0, 6, 26, 52, 0, 1, 0, 0, 0, 36, 150, 203, 0, 1, 0, 0, 0, 24, 150, 962, 877, 0, 1, 0, 0, 0, 0, 240, 900, 6846, 4140, 0, 1, 0, 0, 0, 0, 120, 1560, 9366, 54266, 21147, 0, 1, 0, 0, 0, 0, 0, 1800, 8400, 101556, 471750, 115975, 0
Offset: 0

Views

Author

Seiichi Manyama, Oct 17 2022

Keywords

Examples

			Square array begins:
  1,  1,   1,   1,   1,   1, ...
  0,  1,   0,   0,   0,   0, ...
  0,  2,   2,   0,   0,   0, ...
  0,  5,   6,   6,   0,   0, ...
  0, 15,  26,  36,  24,   0, ...
  0, 52, 150, 150, 240, 120, ...

Links

Crossrefs

Columns k=0-4 give: A000007, A000110, A052859, A353664, A353665.
Cf. A324162, A357293, A357868.

Programs

  • PARI
    T(n, k) = sum(j=0, n, (k*j)!*stirling(n, k*j, 2)/j!);
  • PARI
    T(n, k) = if(k==0, 0^n, n!*polcoef(exp((exp(x+x*O(x^n))-1)^k), n));

Formula

For k > 0, e.g.f. of column k: exp((exp(x) - 1)^k).
T(0,k) = 1; T(n,k) = k! * Sum_{j=1..n} binomial(n-1,j-1) * Stirling2(j,k) * T(n-j,k).

A357010 E.g.f. satisfies log(A(x)) = (exp(x) - 1)^3 * A(x).

Original entry on oeis.org

1, 0, 0, 6, 36, 150, 1620, 24486, 293076, 3843510, 68254740, 1311687366, 25479935316, 552545882070, 13437670215060, 345157499363046, 9370414233900756, 274413997443811830, 8572526271218671380, 281754864204797848326, 9767868351458229261396
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2022

Keywords

Links

Crossrefs

Cf. A052880, A357009.
Cf. A353664, A357025.

Programs

  • Mathematica
    nmax = 20; A[_] = 1;
Do[A[x_] = Exp[(Exp[x] - 1)^3*A[x]] + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 05 2024 *)
  • PARI
    a(n) = sum(k=0, n\3, (3*k)!*(k+1)^(k-1)*stirling(n, 3*k, 2)/k!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(exp(x)-1)^(3*k)/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-(exp(x)-1)^3))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-lambertw(-(exp(x)-1)^3)/(exp(x)-1)^3))

Formula

a(n) = Sum_{k=0..floor(n/3)} (3*k)! * (k+1)^(k-1) * Stirling2(n,3*k)/k!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (exp(x) - 1)^(3*k) / k!.
E.g.f.: A(x) = exp( -LambertW(-(exp(x) - 1)^3) ).
E.g.f.: A(x) = -LambertW(-(exp(x) - 1)^3)/(exp(x) - 1)^3.
a(n) ~ sqrt(1 + exp(1/3)) * 3^n * n^(n-1) / (exp(n-1) * (3*log(1 + exp(1/3)) - 1)^(n - 1/2)). - Vaclav Kotesovec, Sep 27 2023

A375773 Expansion of e.g.f. exp((exp(x) - 1)^5).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 1800, 16800, 126000, 834120, 6917400, 129399600, 3259080000, 72252300120, 1370602233000, 23218349918400, 377834084082000, 6709735404918120, 147369456297228600, 3899127761438053200, 109421543771265852000, 3002806840023201408120
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2024

Keywords

Crossrefs

Cf. A000110, A052859, A353664, A353665.
Cf. A353404, A373940.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((exp(x)-1)^5)))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=120*sum(j=1, i, binomial(i-1, j-1)*stirling(j, 5, 2)*v[i-j+1])); v;
  • PARI
    a(n) = sum(k=0, n\5, (5*k)!*stirling(n, 5*k, 2)/k!);

Formula

G.f.: Sum_{k>=0} (5*k)! * x^(5*k)/(k! * Product_{j=1..5*k} (1 - j * x)).
a(0) = 1; a(n) = 120 * Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,5) * a(n-k).
a(n) = Sum_{k=0..floor(n/5)} (5*k)! * Stirling2(n,5*k)/k!.
Showing 1-7 of 7 results.

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