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.

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)!.

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)!).

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

A353998 Expansion of e.g.f. 1/(1 - x^2/2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 3, 6, 10, 195, 1281, 5908, 68076, 758565, 6486535, 75598446, 1059484218, 13378016743, 185273328345, 2999003869800, 48665352612376, 816394913567433, 15110162148144267, 292156921946387170, 5805684093139498470, 122617308231635240331
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Crossrefs

Cf. A052848, A353999.
Cf. A351505, A354000.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x^2/2*(exp(x)-1))))
  • 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!*stirling(n-2*k, k, 2)/(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)! = binomial(n,2) * Sum_{k=3..n} binomial(n-2,k-2) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/3)} k! * Stirling2(n-2*k,k)/(2^k * (n-2*k)!).
a(n) ~ 2 * n! / ((4 + 2*r + r^3) * r^n), where r = 1.043121496712693605897520269472163423276582653660720448... is the root of the equation (exp(r)-1)*r^2 = 2. - Vaclav Kotesovec, May 13 2022

A368165 Expansion of e.g.f. -log(1 + x^2/2 * log(1 - x)).

Original entry on oeis.org

0, 0, 0, 3, 6, 20, 180, 1134, 7980, 78840, 798840, 8620920, 107668440, 1449377280, 20755871136, 323448048000, 5398086002400, 95487623038080, 1796842848654720, 35808112038746880, 751616958775939200, 16600116241063514880, 384905905873078867200
Offset: 0

Views

Author

Seiichi Manyama, Dec 14 2023

Keywords

Links

Crossrefs

Cf. A052804, A368166.
Cf. A351505.

Programs

  • Mathematica
    With[{nn =30},CoefficientList[Series[-Log[1+x^2/2 Log[1-x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 22 2024 *)
  • PARI
    a(n) = n!*sum(k=1, n\3, (k-1)!*abs(stirling(n-2*k, k, 1))/(2^k*(n-2*k)!));

Formula

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

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)!).

A370996 Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 + x^2/2*log(1-x)) ).

Original entry on oeis.org

1, 0, 0, 3, 6, 20, 810, 6174, 49560, 1439640, 22060080, 312487560, 8687891520, 199853503200, 4216976539776, 126706600944000, 3771722349158400, 106462579493088000, 3626324277349651200, 129806833608095575680, 4565069619653632320000
Offset: 0

Views

Author

Seiichi Manyama, Mar 06 2024

Keywords

Links

Crossrefs

Cf. A370993, A370997.
Cf. A351505.

Programs

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

Formula

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

A375685 Expansion of e.g.f. 1 / (1 + x^2/2 * log(1 - x))^2.

Original entry on oeis.org

1, 0, 0, 6, 12, 40, 720, 4788, 34440, 460080, 5246640, 60318720, 879523920, 13298126400, 206628117696, 3575354428800, 65828785276800, 1264510188264960, 25912058505776640, 561351949518931200, 12721171715573529600, 302794615563937781760, 7554095183751745305600
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Cf. A351505, A375686.
Cf. A375237.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A351505.
a(n) = n! * Sum_{k=0..floor(n/3)} (k+1)! * |Stirling1(n-2*k,k)|/(2^k*(n-2*k)!).

A375686 Expansion of e.g.f. 1 / (1 + x^2/2 * log(1 - x))^3.

Original entry on oeis.org

1, 0, 0, 9, 18, 60, 1350, 9072, 65520, 984960, 11627280, 135883440, 2109317760, 33214821120, 529403146272, 9536973415200, 182108114697600, 3599078480524800, 76130266179974400, 1701744508586747520, 39652022068801632000, 970411293528131750400
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Cf. A351505, A375685.
Cf. A375682.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A351505.
a(n) = (n!/2) * Sum_{k=0..floor(n/3)} (k+2)! * |Stirling1(n-2*k,k)|/(2^k*(n-2*k)!).

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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