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 1-10 of 10 results.

A351506 Expansion of e.g.f. 1/(1 + x^3/6 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 2464, 20160, 178800, 1755600, 21215040, 268107840, 3596916960, 51452200800, 800489733120, 13262804755200, 232536822336000, 4300843392518400, 84023034413644800, 1727339274045504000, 37248117171719731200, 840387048760633651200
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351505.
Cf. A351493, A351504,

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+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!/6*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))/(6^k*(n-3*k)!));

Formula

a(0) = 1; a(n) = n!/6 * 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)|/(6^k * (n-3*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

A354001 Expansion of e.g.f. exp(x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 616, 5124, 29520, 138765, 942700, 9369646, 91711984, 782281955, 6539493520, 62576274440, 693828386976, 7968383514969, 89851862221140, 1023732374445970, 12384993316732960, 160496534000858671, 2163244034675904664, 29653387436468336300
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Links

Crossrefs

Cf. A052506, A354000.
Cf. A351493, A353999.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[x^3/6 (Exp[x]-1)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 07 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/6*(exp(x)-1))))
  • 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, stirling(n-3*k, k, 2)/(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)} Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A355610 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/k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 2, 6, 1, 0, 0, 3, 24, 1, 0, 0, 3, 20, 120, 1, 0, 0, 0, 6, 90, 720, 1, 0, 0, 0, 4, 20, 594, 5040, 1, 0, 0, 0, 0, 10, 180, 4200, 40320, 1, 0, 0, 0, 0, 5, 40, 1134, 34544, 362880, 1, 0, 0, 0, 0, 0, 15, 210, 7980, 316008, 3628800, 1, 0, 0, 0, 0, 0, 6, 70, 1904, 71280, 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,   3,  0,  0, 0, 0, ...
   24,  20,   6,  4,  0, 0, 0, ...
  120,  90,  20, 10,  5, 0, 0, ...
  720, 594, 180, 40, 15, 6, 0, ...

Links

Crossrefs

Columns k=0..4 give A000142, A066166, A351492, A351493, A355507.
Cf. A355609, A355619.

Programs

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

Formula

T(0,k) = 1 and T(n,k) = (n-1)!/k! * 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)|/(k!^j * (n-k*j)!).

A356753 E.g.f. satisfies A(x) = 1/(1 - x)^(x^3/6 * A(x)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 3024, 25200, 225000, 2217600, 29974560, 400720320, 5558957040, 81340459200, 1344965825280, 23566775232000, 432681781459200, 8309927446329600, 170258024427580800, 3679448236206220800, 83235946152090547200, 1962840630226968307200
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A351493, A356913.
Cf. A355842, A356752.

Programs

  • Mathematica
    nmax = 23; A[_] = 1;
Do[A[x_] = 1/(1 - x)^(x^3/6*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))/(6^k*(n-3*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-x^3/6*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(x^3/6*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(lambertw(x^3/6*log(1-x))/(x^3/6*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
E.g.f.: A(x) = Sum_{k>=0} (k+1)^(k-1) * (-x^3/6 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( -LambertW(x^3/6 * log(1-x)) ).
E.g.f.: A(x) = LambertW(x^3/6 * log(1-x))/(x^3/6 * log(1-x)).

A355619 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/k!).

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, -3, 0, 1, 0, 0, 3, 20, 0, 1, 0, 0, 0, -6, -90, 0, 1, 0, 0, 0, 4, 20, 594, 0, 1, 0, 0, 0, 0, -10, 0, -4200, 0, 1, 0, 0, 0, 0, 5, 40, -126, 34544, 0, 1, 0, 0, 0, 0, 0, -15, -210, 1260, -316008, 0, 1, 0, 0, 0, 0, 0, 6, 70, 1904, -4320, 3207240, 0
Offset: 0

Views

Author

Seiichi Manyama, Jul 10 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,  3,   0,   0, 0, 0, ...
  0,  20, -6,   4,   0, 0, 0, ...
  0, -90, 20, -10,   5, 0, 0, ...
  0, 594,  0,  40, -15, 6, 0, ...

Links

Crossrefs

Columns k=1..4 give A007113, A355605, (-1)^n * A351493(n), A355603.
Cf. A355607, A355610.

Programs

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

Formula

T(0,k) = 1 and T(n,k) = -(n-1)!/k! * 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)/(k!^j * (n-k*j)!).

A368166 Expansion of e.g.f. -log(1 + x^3/6 * log(1 - x)).

Original entry on oeis.org

0, 0, 0, 0, 4, 10, 40, 210, 1904, 15120, 132600, 1293600, 14673120, 178738560, 2341182480, 32915282400, 499117301760, 8075042976000, 138689356915200, 2519863488979200, 48354005826489600, 976893364144857600, 20721305503846886400, 460363370406207206400
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Comments

This sequence is different from A351493.

Links

Crossrefs

Cf. A052804, A368165.
Cf. A351493, A351506.

Programs

  • PARI
    a(n) = n!*sum(k=1, n\4, (k-1)!*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=1..floor(n/4)} (k-1)! * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
a(0) = a(1) = a(2) = a(3) = 0; a(n) = n!/(6*(n-3)) + Sum_{k=4..n-1} k!/(6*(k-3)) * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Jan 22 2025

A356913 E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^(x^3/6).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 784, 5040, 40200, 369600, 5285280, 72072000, 1006889520, 14760345600, 210510263040, 3131345817600, 49229619129600, 818940523564800, 15054020163619200, 301204611031564800, 6455999452413772800, 146587705490513548800
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2022

Keywords

Links

Crossrefs

Cf. A356905, A356912.
Cf. A351493, A356753.

Programs

  • Mathematica
    nmax = 23; A[_] = 1;
Do[A[x_] = ((1 - x)^(-x^3/6))^(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))/(6^k*(n-3*k)!));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-x^3/6*log(1-x))^k/k!)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-x^3/6*log(1-x)))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-x^3/6*log(1-x)/lambertw(-x^3/6*log(1-x))))

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (-k+1)^(k-1) * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
E.g.f.: A(x) = Sum_{k>=0} (-k+1)^(k-1) * (-x^3/6 * log(1-x))^k / k!.
E.g.f.: A(x) = exp( LambertW(-x^3/6 * log(1-x)) ).
E.g.f.: A(x) = -x^3/6 * log(1-x)/LambertW(-x^3/6 * log(1-x)).

A355507 Expansion of e.g.f. (1 - x)^(-x^4/24).

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 15, 70, 420, 3024, 28350, 272250, 2875950, 33333300, 420840420, 5763671550, 84799915200, 1334007397800, 22343877115560, 396971840865600, 7456250728017000, 147612122975772000, 3071792315894841000, 67030983483724953000, 1530448652869851191400
Offset: 0

Views

Author

Seiichi Manyama, Jul 09 2022

Keywords

Crossrefs

Column k=4 of A355610.
Cf. A351493.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace((1-x)^(-x^4/24)))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x^4/24*log(1-x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*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))/(24^k*(n-4*k)!));

Formula

a(0) = 1; a(n) = (n-1)!/24 * 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)|/(24^k * (n-4*k)!).
a(n) ~ n! / (Gamma(1/24) * n^(23/24)). - Vaclav Kotesovec, Jul 21 2022
Showing 1-10 of 10 results.

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