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 11 results. Next

A347015 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(1/3).

Original entry on oeis.org

1, 1, 5, 42, 498, 7644, 144156, 3225648, 83536008, 2457701928, 80970232104, 2953056534768, 118112744060208, 5140622709134496, 241863782829704928, 12232551538417012992, 661818290353375962240, 38140594162828447248000, 2332567001993176540206720, 150880256846462633823648000
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 10 2021

Keywords

Links

Crossrefs

Cf. A007559, A007840, A346978, A346982, A347016.
Cf. A354263.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end:
a:= n-> add(abs(Stirling1(n, k))*g(k), k=0..n):
seq(a(n), n=0..19);  # Alois P. Heinz, Aug 10 2021
  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 + 3 Log[1 - x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Abs[StirlingS1[n, k]] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}]

Formula

a(n) = Sum_{k=0..n} |Stirling1(n,k)| * A007559(k).
a(n) ~ n! * exp(n/3) / (Gamma(1/3) * 3^(1/3) * n^(2/3) * (exp(1/3) - 1)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023

A365575 Expansion of e.g.f. 1 / (1 + 3 * log(1-x))^(2/3).

Original entry on oeis.org

1, 2, 12, 114, 1482, 24468, 490020, 11538840, 312363720, 9556741440, 326076452640, 12275391192480, 505400508041760, 22590511357965120, 1089423938332883520, 56379459359942190720, 3116574045158647605120, 183271869976364873222400
Offset: 0

Views

Author

Seiichi Manyama, Sep 09 2023

Keywords

Crossrefs

Cf. A347015, A354263.

Programs

  • Mathematica
    a[n_] := Sum[Product[3*j + 2, {j, 0, k - 1}] * Abs[StirlingS1[n, k]], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 3*j+2)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j+2)) * |Stirling1(n,k)|.
a(0) = 1; a(n) = Sum_{k=1..n} (3 - k/n) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ Gamma(1/3) * n^(n + 1/6) / (3^(1/6) * sqrt(2*Pi) * (exp(1/3) - 1)^(n + 2/3) * exp(2*n/3)). - Vaclav Kotesovec, Nov 11 2023

A320079 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. 1/(1 + k*log(1 - x)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 14, 0, 1, 4, 21, 76, 88, 0, 1, 5, 36, 222, 772, 694, 0, 1, 6, 55, 488, 3132, 9808, 6578, 0, 1, 7, 78, 910, 8824, 55242, 149552, 72792, 0, 1, 8, 105, 1524, 20080, 199456, 1169262, 2660544, 920904, 0, 1, 9, 136, 2366, 39708, 553870, 5410208, 28873800, 54093696, 13109088, 0
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 05 2018

Keywords

Examples

			E.g.f. of column k: A_k(x) = 1 + k*x/1! + k*(2*k + 1)*x^2/2! + 2*k*(3*k^2 + 3*k + 1)*x^3/3! + 2*k*(12*k^3 + 18*k^2 + 11*k + 3)*x^4/4! + ...
Square array begins:
  1,    1,     1,      1,       1,       1,  ...
  0,    1,     2,      3,       4,       5,  ...
  0,    3,    10,     21,      36,      55,  ...
  0,   14,    76,    222,     488,     910,  ...
  0,   88,   772,   3132,    8824,   20080,  ...
  0,  694,  9808,  55242,  199456,  553870,  ...

Crossrefs

Columns k=0..5 give A000007, A007840, A088500, A354263, A354264, A365588.
Main diagonal gives A317171.
Cf. A048594, A048994, A094416, A320080.

Programs

  • Mathematica
    Table[Function[k, n! SeriesCoefficient[1/(1 + k Log[1 - x]), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten

Formula

E.g.f. of column k: 1/(1 + k*log(1 - x)).
A(n,k) = Sum_{j=0..n} |Stirling1(n,j)|*j!*k^j.
A(0,k) = 1; A(n,k) = k * Sum_{j=1..n} (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A375946 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(4/3).

Original entry on oeis.org

1, 4, 32, 372, 5652, 105936, 2360712, 60956472, 1789413864, 58850914752, 2143354213728, 85629122177760, 3723269780412000, 175035687610956480, 8846458578801144000, 478330017277120767360, 27551501517174431852160, 1684176901225092936990720
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Cf. A052801, A375945.
Cf. A354263, A365575, A375951.
Cf. A007559.

Programs

  • Mathematica
    nmax=17; CoefficientList[Series[1 / (1 + 3 * Log[1-x])^(4/3),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a007559(n) = prod(k=0, n-1, 3*k+1);
a(n) = sum(k=0, n, a007559(k+1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} A007559(k+1) * |Stirling1(n,k)|.

A354287 Expansion of e.g.f. 1/(1 - x)^(3/(1 + 3 * log(1-x))).

Original entry on oeis.org

1, 3, 30, 438, 8334, 194580, 5368662, 170591022, 6126386724, 245127214548, 10804866210648, 519910458588576, 27105081897342816, 1521393008601586536, 91445577404393807928, 5858664681621903625368, 398467273528657973600208, 28668189882264862351707504
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A088815, A354286.
Cf. A000262, A354263, A354289, A354291.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^(3/(1+3*log(1-x)))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 3^k*k!*abs(stirling(j, k, 1)))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354263(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * A000262(k) * |Stirling1(n,k)|.
a(n) ~ exp((-5 + 1/(exp(1/3) - 1) + 4*sqrt(3*n/(exp(1/3) - 1)) - 4*n)/6) * n^(n - 1/4) / (sqrt(2) * 3^(1/4) * (exp(1/3) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

A354750 Expansion of e.g.f. 1 / (1 - log(1 + 3*x) / 3).

Original entry on oeis.org

1, 1, -1, 6, -48, 534, -7542, 129240, -2603736, 60292512, -1577546928, 46021512096, -1480976147664, 52110720451152, -1990258155061776, 81995762243700864, -3624527727510038784, 171109526616468957312, -8591991935936929932672, 457246520477143117555968
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2022

Keywords

Crossrefs

Cf. A006252, A087674, A255927, A335531, A352069, A354237, A354263, A354751.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(1 - Log[1 + 3 x]/3), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 3^(n - k), {k, 0, n}], {n, 0, 19}]
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+3*x)/3))) \\ Michel Marcus, Jun 06 2022

Formula

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

A375951 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^(5/3).

Original entry on oeis.org

1, 5, 45, 570, 9270, 183840, 4299360, 115795920, 3528915840, 120032889840, 4507313333040, 185185602462240, 8262852630732000, 397873645339668480, 20563762111640910720, 1135441077379757372160, 66703342626913255770240, 4154100873615633462894720
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Cf. A354263, A365575, A375946.
Cf. A008544.

Programs

  • Mathematica
    nmax=17; CoefficientList[Series[1 / (1 + 3 * Log[1-x])^(5/3),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a008544(n) = prod(k=0, n-1, 3*k+2);
a(n) = sum(k=0, n, a008544(k+1)*abs(stirling(n, k, 1)))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} A008544(k+1) * |Stirling1(n,k)|.

A375988 Expansion of e.g.f. (1 + 3 * log(1 - x))^(4/3).

Original entry on oeis.org

1, -4, 0, 12, 108, 1104, 14136, 225768, 4386168, 100885248, 2683789344, 81047258208, 2737919298528, 102266990392896, 4184016413001408, 186047367206499072, 8933002185371731200, 460580247564830138880, 25378595790818821816320, 1488230641037882346324480
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Cf. A354263, A365575, A375946, A375951, A375989.

Programs

  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 3*j-4)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (3*j-4)) * |Stirling1(n,k)|.

A375721 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^2.

Original entry on oeis.org

1, 6, 60, 822, 14238, 297684, 7286076, 204251328, 6450932448, 226613038608, 8763294140064, 369900822475728, 16922169163019088, 833991953707934496, 44050579327333028448, 2482381132145285334912, 148660444826262311114880, 9427874254540824544312320
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A354263, A375722.
Cf. A052801, A367474.
Cf. A367472.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))^2))
  • PARI
    a(n) = sum(k=0, n, 3^k*(k+1)!*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} 3^k * (k+1)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * exp(2*n/3) * (exp(1/3) - 1)^(n+2)). - Vaclav Kotesovec, Sep 06 2024

A375722 Expansion of e.g.f. 1 / (1 + 3 * log(1 - x))^3.

Original entry on oeis.org

1, 9, 117, 1962, 40122, 966276, 26755812, 836862192, 29167596504, 1120629465432, 47044646845848, 2142210019297680, 105154320625284240, 5534780654854980000, 310945503593770489440, 18570787974013838515200, 1174884522886771261079040
Offset: 0

Views

Author

Seiichi Manyama, Aug 25 2024

Keywords

Crossrefs

Cf. A354263, A375721.
Cf. A354122, A367475.
Cf. A367473.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1+3*log(1-x))^3))
  • PARI
    a(n) = sum(k=0, n, 3^k*(k+2)!*abs(stirling(n, k, 1)))/2;

Formula

a(n) = (1/2) * Sum_{k=0..n} 3^k * (k+2)! * |Stirling1(n,k)|.
a(0) = 1; a(n) = 3 * Sum_{k=1..n} (2*k/n + 1) * (k-1)! * binomial(n,k) * a(n-k).
Showing 1-10 of 11 results. Next

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

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