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.

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)!.

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

A358013 Expansion of e.g.f. 1/(1 - x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 750, 5082, 23576, 453672, 5755770, 50894030, 841270452, 14694142476, 201442729670, 3552604015170, 73814245552560, 1369932831933392, 27860865121662066, 655240785723048726, 15052226249248287500, 357713461766745539700, 9416426612423343023742
Offset: 0

Views

Author

Seiichi Manyama, Oct 24 2022

Keywords

Crossrefs

Cf. A052848, A358014.
Cf. A240989, A351503, A353998.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2*(exp(x)-1))))
  • 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!*stirling(n-2*k, k, 2)/(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! * Stirling2(n-2*k,k)/(n-2*k)!.

A370994 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^2*log(1-x)) ).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 3060, 23688, 191520, 9698400, 158548320, 2304973440, 100716073920, 2627516361600, 58513944513024, 2512156283683200, 89046056086041600, 2739316757454950400, 124170651534918297600, 5440968468533003212800, 215067442349096186572800
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A052802, A370993, A370995.
Cf. A351503, A370989.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1+x^2*log(1-x)))/x))
  • PARI
    a(n) = sum(k=0, n\3, (n+k)!*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/(n+1);

Formula

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

A371302 E.g.f. satisfies A(x) = 1/(1 + x^2*log(1 - x*A(x))).

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 1620, 13608, 117600, 2924640, 49603680, 782147520, 19083936960, 463369645440, 10836652514688, 304533583200000, 9218842256332800, 281872333420554240, 9421579421176089600, 338543319734116116480, 12590519274541116518400
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Crossrefs

Cf. A370994, A371118, A371138.
Cf. A351503.

Programs

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

Formula

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

A375639 Expansion of e.g.f. 1 / (1 + x^2 * log(1 - x))^2.

Original entry on oeis.org

1, 0, 0, 12, 24, 80, 2520, 17136, 124320, 2462400, 30965760, 372113280, 7014807360, 122840789760, 2078973921024, 43236813312000, 932206147891200, 20090534745415680, 480054835899371520, 12126262777282805760, 313198020852233932800
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Cf. A351503, A375679.
Cf. A375662.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1+x^2 Log[1-x])^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 29 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^2*log(1-x))^2))
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+1)!*abs(stirling(n-2*k, k, 1))/(n-2*k)!);

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A351503.
a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A375679 Expansion of e.g.f. 1 / (1 + x^2 * log(1 - x))^3.

Original entry on oeis.org

1, 0, 0, 18, 36, 120, 4860, 33264, 241920, 5598720, 72364320, 879500160, 18172978560, 331463508480, 5726430597888, 126134466796800, 2836325702246400, 62773403361177600, 1562890149787392000, 41009994647421972480, 1090182759179092992000
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Cf. A351503, A375639.
Cf. A375663.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x^2*log(1-x))^3))
  • PARI
    a(n) = n!*sum(k=0, n\3, (k+2)!*abs(stirling(n-2*k, k, 1))/(n-2*k)!)/2;

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A351503.
a(n) = (n!/2) * Sum_{k=0..floor(n/3)} (k+2)! * |Stirling1(n-2*k,k)|/(n-2*k)!.

A355665 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 * 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, 6, 32, 694, 1, 0, 0, 0, 12, 150, 6578, 1, 0, 0, 0, 24, 40, 1524, 72792, 1, 0, 0, 0, 0, 60, 900, 12600, 920904, 1, 0, 0, 0, 0, 120, 240, 6048, 147328, 13109088, 1, 0, 0, 0, 0, 0, 360, 1260, 43680, 1705536, 207360912
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,   6,   0,   0,   0, 0, ...
    88,   32,  12,  24,   0,   0, 0, ...
   694,  150,  40,  60, 120,   0, 0, ...
  6578, 1524, 900, 240, 360, 720, 0, ...

Crossrefs

Columns k=0..3 give A007840, A052830, A351503, A351504.
Cf. A355609, A355652.

Programs

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

Formula

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

A375698 Expansion of e.g.f. 1 / sqrt(1 + x^2 * log(1 - x)).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 360, 2394, 17220, 252720, 2963520, 34525260, 552027960, 8860952880, 142907532768, 2682870913800, 53297669552400, 1086135012144000, 24087251436249600, 566843973576536880, 13834256829134364000, 357412359616922433600, 9723652519748883408000
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A001147, A351503, A351505, A375715.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/sqrt(1+x^2*log(1-x))))
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = n!*sum(k=0, n\3, a001147(k)*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} A001147(k) * |Stirling1(n-2*k,k)|/(2^k*(n-2*k)!).
Showing 1-10 of 10 results.

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