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.

A351503 Expansion of e.g.f. 1/(1 + x^2 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 900, 6048, 43680, 717120, 8658720, 102231360, 1735525440, 28819964160, 473955850368, 9235543363200, 189202617676800, 3940225003653120, 89804740509434880, 2169337606086389760, 54085753764912844800, 1429100881569205125120
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351504.
Cf. A351505, A353228.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[1/(1+x^2 Log[1-x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 18 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+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!*sum(j=3, i, 1/(j-2)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\3, k!*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

a(0) = 1; a(n) = n! * Sum_{k=3..n} 1/(k-2) * a(n-k)/(n-k)!.
a(n) = n! * Sum_{k=0..floor(n/3)} k! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A353229 Expansion of e.g.f. (1 - x)^(-x^3).

Original entry on oeis.org

1, 0, 0, 0, 24, 60, 240, 1260, 28224, 241920, 2181600, 21621600, 315342720, 4358914560, 61607407680, 912518006400, 15142006978560, 265601118182400, 4877947688140800, 93691850626483200, 1901787789077452800, 40548028309147699200, 904101131200045363200
Offset: 0

Views

Author

Seiichi Manyama, May 01 2022

Keywords

Links

Crossrefs

Cf. A066166, A351493, A351504, A353228.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1-x)^-x^3,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 20 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^3)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-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-1)!*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))/(n-3*k)!);

Formula

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

A351492 Expansion of e.g.f. (1 - x)^(-x^2/2).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 180, 1134, 7980, 71280, 685440, 7165620, 82720440, 1036404720, 13990472496, 202812132600, 3141926096400, 51795939162240, 905465629762560, 16731527824735920, 325859956191352800, 6671593966263992640, 143254214818174152000
Offset: 0

Views

Author

Seiichi Manyama, May 02 2022

Keywords

Links

Crossrefs

Cf. A351493, A351505, A353228.

Programs

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

Formula

a(0) = 1; a(n) = ((n-1)!/2) * 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)|/(2^k * (n-2*k)!).
a(n) ~ sqrt(2) * n^n / exp(n). - Vaclav Kotesovec, May 04 2022

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)!.

A356910 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^2).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, -180, -1512, -11760, 142560, 2701440, 37033920, -47472480, -7299227520, -181704466944, -904179830400, 40024286265600, 1774386897454080, 24426730612869120, -217650777809310720, -26326923875473536000, -662608157128469637120
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A356905, A356911.
Cf. A353228, A355287.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^2))^(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\3, (-k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^2*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^2*log(1-x)/lambertw(-x^2*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (-k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^2 * log(1-x)) ).
E.g.f.: A(x) = -x^2 * log(1-x)/LambertW(-x^2 * log(1-x)).

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

A355287 E.g.f. satisfies A(x) = 1/(1 - x)^(x^2 * A(x)).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 1260, 8568, 62160, 1473120, 19111680, 232626240, 5403451680, 103176028800, 1822033204992, 45916616592000, 1129459815993600, 26346457488798720, 749439127417466880, 22165051763204582400, 640916967497214643200, 20787453048015928350720
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A353228, A356910.

Programs

  • Mathematica
    nmax = 21; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^2*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\3, (k+1)^(k-1)*abs(stirling(n-2*k, k, 1))/(n-2*k)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^2*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^2*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^2*log(1-x))/(x^2*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)^(k-1) * |Stirling1(n-2*k,k)|/(n-2*k)!.
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^2 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^2 * log(1-x)) ).
E.g.f.: A(x) = LambertW(x^2 * log(1-x))/(x^2 * log(1-x)).

A362891 Expansion of e.g.f. 1/(1 + LambertW(x^2 * log(1-x))).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 1620, 11088, 80640, 2289600, 30471840, 374663520, 9819817920, 195106129920, 3507260492736, 95860364846400, 2466492401318400, 58909563259223040, 1775000008437557760, 54856736708999339520, 1629826915777548364800
Offset: 0

Views

Author

Seiichi Manyama, May 08 2023

Keywords

Links

Crossrefs

Cf. A305981, A362835.
Cf. A353228, A362892.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+lambertw(x^2*log(1-x)))))

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} k^k * |Stirling1(n-2*k,k)|/(n-2*k)!.
Showing 1-8 of 8 results.

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