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.

A351493 Expansion of e.g.f. (1 - x)^(-x^3/6).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 1904, 15120, 132600, 1293600, 14303520, 171531360, 2223464880, 31023392400, 464541960960, 7424367350400, 126124766476800, 2269425252931200, 43119553374460800, 862673918061715200, 18126931548822835200, 399119899456951411200
Offset: 0

Views

Author

Seiichi Manyama, May 02 2022

Keywords

Links

Crossrefs

Cf. A351492, A351506, A353229.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^3/6)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^3/6*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=4, i, j/(j-3)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = (n-1)!/6 * Sum_{k=4..n} k/(k-3) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
a(n) ~ sqrt(2*Pi) * n^(n - 1/3) / (Gamma(1/6) * exp(n)). - Vaclav Kotesovec, May 04 2022

A351504 Expansion of e.g.f. 1/(1 + x^3 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 48384, 423360, 3844800, 38253600, 896797440, 14322147840, 216997522560, 3350656108800, 74820944056320, 1621271286835200, 34293811249152000, 727304513980262400, 18147791755697356800, 476653146551318016000
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351503.
Cf. A351506, A353229.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^3*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=4, i, 1/(j-3)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\4, k!*abs(stirling(n-3*k, k, 1))/(n-3*k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=4..n} 1/(k-3) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/4)} k! * |Stirling1(n-3*k,k)|/(n-3*k)!.

A353228 Expansion of e.g.f. (1 - x)^(-x^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 540, 3528, 25200, 263520, 2741760, 30048480, 372794400, 4971957120, 70612686144, 1076056027200, 17469796780800, 300562292459520, 5468568356666880, 104917700221125120, 2116572758902425600, 44794683422986936320, 992435268252158438400
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Links

Crossrefs

Cf. A066166, A351503, A351492, A353223, A353229.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1-x)^(-x^2), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, May 12 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^2)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^2*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} |Stirling1(n-2*k,k)|/(n-2*k)!.
a(n) ~ sqrt(2*Pi) * n^(n + 1/2) / exp(n). - Vaclav Kotesovec, May 04 2022

A355607 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 + x)^(x^k).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 6, 20, 0, 1, 0, 0, 0, -12, -90, 0, 1, 0, 0, 0, 24, 40, 594, 0, 1, 0, 0, 0, 0, -60, 180, -4200, 0, 1, 0, 0, 0, 0, 120, 240, -1512, 34544, 0, 1, 0, 0, 0, 0, 0, -360, -1260, 11760, -316008, 0, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, -38880, 3207240, 0
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,   1,    1,   1, 1, ...
  1,   0,   0,   0,    0,   0, 0, ...
  0,   2,   0,   0,    0,   0, 0, ...
  0,  -3,   6,   0,    0,   0, 0, ...
  0,  20, -12,  24,    0,   0, 0, ...
  0, -90,  40, -60,  120,   0, 0, ...
  0, 594, 180, 240, -360, 720, 0, ...

Links

Crossrefs

Columns k=1..4 give A007113, A007121, (-1)^n * A353229(n), A354625.
Cf. A292892, A355609, A355619.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), stirling(n-k*j, j, 1)/(n-k*j)!);

Formula

T(0,k) = 1 and T(n,k) = -(n-1)! * Sum_{j=k+1..n} (-1)^(j-k) * j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} Stirling1(n-k*j,j)/(n-k*j)!.

A355609 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. (1 - x)^(-x^k).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 6, 20, 120, 1, 0, 0, 0, 12, 90, 720, 1, 0, 0, 0, 24, 40, 594, 5040, 1, 0, 0, 0, 0, 60, 540, 4200, 40320, 1, 0, 0, 0, 0, 120, 240, 3528, 34544, 362880, 1, 0, 0, 0, 0, 0, 360, 1260, 25200, 316008, 3628800, 1, 0, 0, 0, 0, 0, 720, 1680, 28224, 263520, 3207240, 39916800
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Examples

			Square array begins:
    1,   1,   1,   1,   1,   1, 1, ...
    1,   0,   0,   0,   0,   0, 0, ...
    2,   2,   0,   0,   0,   0, 0, ...
    6,   3,   6,   0,   0,   0, 0, ...
   24,  20,  12,  24,   0,   0, 0, ...
  120,  90,  40,  60, 120,   0, 0, ...
  720, 594, 540, 240, 360, 720, 0, ...

Links

Crossrefs

Columns k=0..4 give A000142, A066166, A353228, A353229, A354624.
Cf. A355607, A355610.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), abs(stirling(n-k*j, j, 1))/(n-k*j)!);

Formula

T(0,k) = 1 and T(n,k) = (n-1)! * Sum_{j=k+1..n} j/(j-k) * T(n-j,k)/(n-j)! for n > 0.
T(n,k) = n! * Sum_{j=0..floor(n/(k+1))} |Stirling1(n-k*j,j)|/(n-k*j)!.

A354624 Expansion of e.g.f. (1 - x)^(-x^4).

Original entry on oeis.org

1, 0, 0, 0, 0, 120, 360, 1680, 10080, 72576, 2419200, 25660800, 279417600, 3286483200, 41894012160, 794511244800, 13755488947200, 238514695372800, 4269265386946560, 79696849513881600, 1658065431859200000
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Crossrefs

Column k=4 of A355609.
Cf. A353228, A353229.
Cf. A354625.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^4)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^4*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=5, i, j/(j-4)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\5, abs(stirling(n-4*k, k, 1))/(n-4*k)!);

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=5..n} k/(k-4) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/5)} |Stirling1(n-4*k,k)|/(n-4*k)!.
a(n) ~ n! * (1 - 4/n - 16*log(n)/n^2). - Vaclav Kotesovec, Jul 21 2022

A356099 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3 * A(x)).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 68544, 604800, 5508000, 54885600, 1877420160, 32069157120, 499522645440, 7832035411200, 236207887534080, 5868136834560000, 133085307920947200, 2941187195765145600, 91568561750088652800, 2857211689810118860800
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A353229, A356911.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^3*A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^3*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3*log(1-x))/(x^3*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^3 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3 * log(1-x)) ).
E.g.f.: A(x) = LambertW(x^3 * log(1-x))/(x^3 * log(1-x)).

A356911 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, -12096, -120960, -1144800, -11642400, 190270080, 4670265600, 81378198720, 1348668921600, -880532674560, -406217626214400, -13255586359142400, -343166884178227200, -3137937973466572800, 72862796986940620800
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A356905, A356910.
Cf. A353229, A356099.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^3))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (-k+1)^(k-1)*abs(stirling(n-3*k, k, 1))/(n-3*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^3*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3*log(1-x)/lambertw(-x^3*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-k+1)^(k-1) * |Stirling1(n-3*k,k)|/(n-3*k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^3 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^3 * log(1-x)) ).
E.g.f.: A(x) = -x^3 * log(1-x)/LambertW(-x^3 * log(1-x)).
Showing 1-8 of 8 results.

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