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.

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A375561 Expansion of e.g.f. 1 / (1 + x * log(1 - x^2)).

Original entry on oeis.org

1, 0, 0, 6, 0, 60, 720, 1680, 40320, 453600, 3326400, 67858560, 878169600, 11935123200, 240708948480, 3946374432000, 73927190937600, 1621341859737600, 32960791774310400, 758085507686707200, 18570669277095936000, 454016684061997056000, 12100759898595611443200
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052830, A375562.
Cf. A353226.

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, (n-2*k)!*abs(stirling(k, n-2*k, 1))/k!);

Formula

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

A367879 Expansion of e.g.f. 1/(1 + 3 * x * log(1-x)).

Original entry on oeis.org

1, 0, 6, 9, 240, 1170, 25812, 244440, 5464512, 79579584, 1926411120, 37900930320, 1018338863616, 25047229315680, 752077828672128, 22027545026192160, 738063856107279360, 24935406131189352960, 927531711339595204608, 35370336293213512527360
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2023

Keywords

Crossrefs

Cf. A052830, A367878.

Programs

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

Formula

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

A354327 Expansion of e.g.f. 1/(1 + x/4 * log(1 - 2 * x)).

Original entry on oeis.org

1, 0, 1, 3, 22, 180, 1902, 23730, 344872, 5706288, 105960600, 2181449160, 49311653616, 1214109056160, 32339248301808, 926527371653520, 28410493609687680, 928335829570087680, 32201658919855225728, 1181755749910942408320, 45744743939940787150080
Offset: 0

Views

Author

Seiichi Manyama, May 24 2022

Keywords

Crossrefs

Cf. A052830, A187735, A354325, A354328.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x/4*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-3)/(j-1)*v[i-j+1]/(i-j)!)); v;
  • PARI
    a(n) = n!*sum(k=0, n\2, 2^(n-3*k)*k!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

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

A367878 Expansion of e.g.f. 1/(1 + 2 * x * log(1-x)).

Original entry on oeis.org

1, 0, 4, 6, 112, 540, 8688, 78960, 1343872, 18162144, 346968000, 6157134720, 134058110976, 2912224423680, 72152130903552, 1839996238429440, 51471401675489280, 1500206702407741440, 46934038380170391552, 1535198134749947965440
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2023

Keywords

Crossrefs

Cf. A052830, A367879.
Cf. A367883.

Programs

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

Formula

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

A371140 E.g.f. satisfies A(x) = 1 - x*A(x)^2 * log(1 - x).

Original entry on oeis.org

1, 0, 2, 3, 56, 270, 5064, 47040, 984416, 14116032, 336538080, 6589416240, 179336461248, 4446985514400, 137520942168960, 4112410749501600, 143445512622458880, 5004065722611594240, 195260931334478223360, 7762385328551718796800, 336051947630616458065920
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2024

Keywords

Crossrefs

Cf. A052830, A371141.
Cf. A052803.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(-1 + Sqrt[1 + 4*x*Log[1 - x]])/(2*x*Log[1 - x]), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 12 2024 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(2/(1+sqrt(1+4*x*log(1-x)))))
  • PARI
    a(n) = n!*sum(k=0, n\2, (2*k)!/(k+1)!*abs(stirling(n-k, k, 1))/(n-k)!);

Formula

E.g.f.: 2/(1 + sqrt(1+4*x*log(1-x))).
a(n) = n! * Sum_{k=0..floor(n/2)} (2*k)!/(k+1)! * |Stirling1(n-k,k)|/(n-k)!.
a(n) ~ sqrt(2 + 8*r^2/(1-r)) * n^(n-1) / (exp(n) * r^n), where r = 0.436224579489690436773045325306926562580857950193340891933383996... is the root of the equation 4*r*log(1-r) = -1. - Vaclav Kotesovec, Mar 12 2024

A371141 E.g.f. satisfies A(x) = 1 - x*A(x)^3 * log(1 - x).

Original entry on oeis.org

1, 0, 2, 3, 80, 390, 10764, 104160, 3162144, 48889008, 1647798480, 35939566080, 1347110453952, 38272507827840, 1593399505840128, 55860824012535360, 2575479834957911040, 107239963351030433280, 5453101063482843276288, 262319113586136087567360
Offset: 0

Views

Author

Seiichi Manyama, Mar 12 2024

Keywords

Crossrefs

Cf. A052830, A371140.
Cf. A367158.

Programs

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

Formula

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

A375556 Expansion of e.g.f. 1 / (1 + x * log(1 - x^3/6)).

Original entry on oeis.org

1, 0, 0, 0, 4, 0, 0, 70, 1120, 0, 5600, 184800, 2217600, 1201200, 61661600, 1513512000, 16682265600, 38118080000, 1440863424000, 31721866176000, 352561745536000, 2053230379200000, 68832104140800000, 1449890913639168000, 17583390443114496000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052830, A375167, A375558.
Cf. A351156.

Programs

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

Formula

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

A375562 Expansion of e.g.f. 1 / (1 + x * log(1 - x^3)).

Original entry on oeis.org

1, 0, 0, 0, 24, 0, 0, 2520, 40320, 0, 1209600, 39916800, 479001600, 1556755200, 79913433600, 1961511552000, 25107347865600, 296406190080000, 11204153985024000, 263564384219136000, 4284610758844416000, 95795516571955200000, 3345240261242880000000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2024

Keywords

Crossrefs

Cf. A052830, A375561.
Cf. A353227.

Programs

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

Formula

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

A375672 Expansion of e.g.f. 1 / (1 + x * log(1 - x))^3.

Original entry on oeis.org

1, 0, 6, 9, 168, 810, 11592, 103320, 1511808, 19350576, 315908640, 5127930720, 95386497984, 1843728194880, 38978317929600, 866801578406400, 20627303078937600, 516780346452733440, 13695223899883530240, 381043219813390540800, 11135125489382277811200
Offset: 0

Views

Author

Seiichi Manyama, Aug 23 2024

Keywords

Crossrefs

Cf. A052830, A375671.
Cf. A354122, A375661.

Programs

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

Formula

E.g.f.: B(x)^3, where B(x) is the e.g.f. of A052830.
a(n) = (n!/2) * Sum_{k=0..floor(n/2)} (k+2)! * |Stirling1(n-k,k)|/(n-k)!.
Previous Showing 11-20 of 30 results. Next

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

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