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.

Previous Showing 21-30 of 30 results.

A353994 Expansion of e.g.f. 1/(1 + log(1 - x) * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 2, 6, 42, 305, 2815, 29792, 362432, 4952481, 75239143, 1257202584, 22918653428, 452620972245, 9626556838015, 219367419292972, 5332164894151648, 137709755844024929, 3765736630207259055, 108696751776637007080, 3302628833563666988740
Offset: 0

Views

Author

Seiichi Manyama, May 13 2022

Keywords

Links

Crossrefs

Cf. A052830, A052848, A052863, A305407, A353995.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+log(1-x)*(exp(x)-1))))

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A052863(k) * binomial(n,k) * a(n-k).

A354315 Expansion of e.g.f. 1/(1 + x/2 * log(1 - 2 * x)).

Original entry on oeis.org

1, 0, 2, 6, 56, 480, 5664, 75600, 1182208, 20829312, 410768640, 8943010560, 213187497984, 5520777799680, 154333888579584, 4631752470159360, 148523272512307200, 5067610703150284800, 183308248516478828544, 7006773595450681589760, 282194468488468121518080
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052830, A354316.
Cf. A354309, A354327.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x/2*log(1-2*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 2^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-2*k)*k!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

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

A354316 Expansion of e.g.f. 1/(1 + x/3 * log(1 - 3 * x)).

Original entry on oeis.org

1, 0, 2, 9, 96, 1170, 18324, 340200, 7360128, 181476288, 5024611440, 154319988240, 5206240427904, 191372822989920, 7612497915813504, 325791049256094240, 14925809593280332800, 728828735500650355200, 37786217117138333005824
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A052830, A354315.
Cf. A354310.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[1/(1+x/3 Log[1-3x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 06 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x/3*log(1-3*x))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=i!*sum(j=2, i, 3^(j-2)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 3^(n-2*k)*k!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

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

A375558 Expansion of e.g.f. 1 / (1 + x * log(1 - x^4/24)).

Original entry on oeis.org

1, 0, 0, 0, 0, 5, 0, 0, 0, 315, 6300, 0, 0, 150150, 6306300, 94594500, 0, 268017750, 17689171500, 549972423000, 7332965640000, 1283268987000, 117632990475000, 5681673439942500, 155840185781280000, 1961530116170625000, 1606200062942475000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052830, A375167, A375556.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x*log(1-x^4/24))))
  • PARI
    a(n) = n!*sum(k=0, n\4, (n-4*k)!*abs(stirling(k, n-4*k, 1))/(24^k*k!));

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} (n-4*k)! * |Stirling1(k,n-4*k)|/(24^k*k!).

A375671 Expansion of e.g.f. 1 / (1 + x * log(1 - x))^2.

Original entry on oeis.org

1, 0, 4, 6, 88, 420, 5148, 44520, 587424, 7203168, 109106640, 1689621120, 29620245312, 546547098240, 10989238893696, 233884517368320, 5324618721070080, 128058198711690240, 3260308438558826496, 87336328336058603520, 2459915920512955929600
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Cf. A052830, A375672.
Cf. A052801, A375660.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A052830.
a(n) = n! * Sum_{k=0..floor(n/2)} (k+1)! * |Stirling1(n-k,k)|/(n-k)!.

A375684 Expansion of e.g.f. 1 / (1 - x * log(1 - x)).

Original entry on oeis.org

1, 0, -2, -3, 16, 90, -204, -4200, -5312, 254016, 1586160, -17970480, -294932736, 790115040, 54224747136, 216483714720, -10481294822400, -137535688281600, 1798183916660736, 58769251106526720, -95282580797291520, -23811620975395061760, -203282679617698222080
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Cf. A052830, A367878, A367879.
Cf. A375683.

Programs

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

Formula

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

A377438 E.g.f. satisfies A(x) = (1 - x * log(1 - x) * A(x))^2.

Original entry on oeis.org

1, 0, 4, 6, 136, 660, 13668, 128520, 2846240, 41368320, 1021615920, 20260896480, 564541372800, 14159468157120, 445236762450816, 13446791658256320, 474901138629918720, 16708336544212992000, 658279512232521209856, 26360704394322974161920, 1150065728368040063784960
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2024

Keywords

Crossrefs

Cf. A052830, A377691.
Cf. A371140.

Programs

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

Formula

E.g.f.: 4/(1 + sqrt(1 + 4*x*log(1-x)))^2.
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A371140.
a(n) = 2 * n! * Sum_{k=0..floor(n/2)} (2*k+1)! * |Stirling1(n-k,k)|/( (n-k)! * (k+2)! ).

A377691 E.g.f. satisfies A(x) = (1 - x * log(1 - x) * A(x))^3.

Original entry on oeis.org

1, 0, 6, 9, 312, 1530, 47952, 468720, 15273696, 238738752, 8404102080, 185234979600, 7145001364608, 204957002147040, 8705298805015680, 307822476591957600, 14400927608439260160, 604208707715034777600, 31065769175985079142400, 1504405685073556864627200
Offset: 0

Views

Author

Seiichi Manyama, Nov 04 2024

Keywords

Crossrefs

Cf. A052830, A377438.
Cf. A371141.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A371141.
a(n) = 3 * n! * Sum_{k=0..floor(n/2)} (3*k+2)! * |Stirling1(n-k,k)|/( (n-k)! * (2*k+3)! ).

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)!.

A375687 Expansion of e.g.f. 1 / sqrt(1 + 2 * x * log(1 - x)).

Original entry on oeis.org

1, 0, 2, 3, 44, 210, 2934, 26040, 404592, 5302584, 95029560, 1632252600, 33865401096, 712672337520, 16986980278800, 420485947572600, 11386595338156800, 322890555922925760, 9820815078397642560, 313247186941438569600, 10588974153880701225600
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Cf. A001147, A052830, A346978, A367878.

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} A001147(k) * |Stirling1(n-k,k)|/(n-k)!.
Previous Showing 21-30 of 30 results.

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

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