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

A351505 Expansion of e.g.f. 1/(1 + x^2/2 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 270, 1764, 12600, 146880, 1597680, 17934840, 243777600, 3506518080, 52696595952, 870564618000, 15354480960000, 284780747946240, 5622461683666560, 117425971162442880, 2574172644658272000, 59302473667128599040, 1432738540209781728000
Offset: 0

Views

Author

Seiichi Manyama, May 04 2022

Keywords

Links

Crossrefs

Cf. A052830, A351506.
Cf. A351492, A351503,

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+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!/2*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))/(2^k*(n-2*k)!));

Formula

a(0) = 1; a(n) = n!/2 * 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)|/(2^k * (n-2*k)!).

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)!.

A353999 Expansion of e.g.f. 1/(1 - x^3/6 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 20, 35, 1176, 10164, 58920, 277365, 3363580, 47567806, 519759604, 4591587455, 51017687280, 786120055400, 12187597925136, 165128862881769, 2261843835692340, 36940778814100210, 678763188831800380, 12143893591131411571, 211404290379223149384
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Crossrefs

Cf. A052848, A353998.
Cf. A000292, A351506, A354001.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-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!/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!*stirling(n-3*k, k, 2)/(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)! = binomial(n,3) * Sum_{k=4..n} binomial(n-3,k-3) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/4)} k! * Stirling2(n-3*k,k)/(6^k * (n-3*k)!).

A355652 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + x^k/k! * log(1 - x)).

Original entry on oeis.org

1, 1, 1, 1, 0, 3, 1, 0, 2, 14, 1, 0, 0, 3, 88, 1, 0, 0, 3, 32, 694, 1, 0, 0, 0, 6, 150, 6578, 1, 0, 0, 0, 4, 20, 1524, 72792, 1, 0, 0, 0, 0, 10, 270, 12600, 920904, 1, 0, 0, 0, 0, 5, 40, 1764, 147328, 13109088, 1, 0, 0, 0, 0, 0, 15, 210, 12600, 1705536, 207360912, 1, 0, 0, 0, 0, 0, 6, 70, 2464, 146880, 23681520, 3608233056
Offset: 0

Views

Author

Seiichi Manyama, Jul 13 2022

Keywords

Examples

			Square array begins:
     1,    1,   1,  1,  1, 1, 1, ...
     1,    0,   0,  0,  0, 0, 0, ...
     3,    2,   0,  0,  0, 0, 0, ...
    14,    3,   3,  0,  0, 0, 0, ...
    88,   32,   6,  4,  0, 0, 0, ...
   694,  150,  20, 10,  5, 0, 0, ...
  6578, 1524, 270, 40, 15, 6, 0, ...

Links

Crossrefs

Columns k=0..3 give A007840, A052830, A351505, A351506.
Cf. A355610, A355665.

Programs

  • Mathematica
    T[n_, k_] := n! * Sum[j! * Abs[StirlingS1[n - k*j, j]]/(k!^j*(n - k*j)!), {j, 0, Floor[n/(k + 1)]}]; Table[T[k, n - k], {n, 0, 11}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 13 2022 *)
  • PARI
    T(n, k) = n!*sum(j=0, n\(k+1), j!*abs(stirling(n-k*j, j, 1))/(k!^j*(n-k*j)!));

Formula

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

A375699 Expansion of e.g.f. 1 / (1 + x^3 * log(1 - x))^(1/6).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 5264, 45360, 409800, 4065600, 77948640, 1183422240, 17527233360, 267109642800, 5422495921920, 110998923235200, 2270809072896000, 47142009514454400, 1116394268619772800, 27963045712157472000, 718066383283082803200
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A351504, A375700, A375701.
Cf. A008542, A351506.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^3*log(1-x))^(1/6)))
  • PARI
    a(n) = n!*sum(k=0, n\4, prod(j=0, k-1, 6*j+1)*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (Product_{j=0..k-1} (6*j+1)) * |Stirling1(n-3*k,k)|/(6^k*(n-3k)!).

A370997 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^3/6*log(1-x)) ).

Original entry on oeis.org

1, 0, 0, 0, 4, 10, 40, 210, 6944, 65520, 640800, 6837600, 157375680, 2741618880, 45897895680, 783559576800, 18503310228480, 440531086195200, 10407471103411200, 247739364392083200, 6801330820818124800, 198670207398879283200, 5945924796494183424000
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A370993, A370996.
Cf. A351506.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x^3/6*log(1-x)))/x))
  • PARI
    a(n) = sum(k=0, n\4, (n+k)!*abs(stirling(n-3*k, k, 1))/(6^k*(n-3*k)!))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/4)} (n+k)! * |Stirling1(n-3*k,k)|/(6^k * (n-3*k)!).
Showing 1-8 of 8 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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