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 1-9 of 9 results.

A088500 Expansion of e.g.f. 1/(1+2*log(1-x)).

Original entry on oeis.org

1, 2, 10, 76, 772, 9808, 149552, 2660544, 54093696, 1237306560, 31446049728, 879119219328, 26811313164672, 885830291432448, 31518653868782592, 1201567079771092992, 48860409899753588736, 2111033523652100407296
Offset: 0

Views

Author

Vladeta Jovovic, Nov 12 2003

Keywords

Links

Crossrefs

Cf. A007840, A088501.

Programs

  • Mathematica
    CoefficientList[Series[1/(1+2*Log[1-x]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, May 03 2015 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(1/(1+2*log(1-x)))) \\ Michel Marcus, Apr 26 2021

Formula

a(n) = Sum_{k=0..n} |Stirling1(n, k)|*k!*2^k.
a(n) ~ n! * exp(n/2) / (2 * (exp(1/2)-1)^(n+1)). - Vaclav Kotesovec, May 03 2015
a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k). - Ilya Gutkovskiy, Apr 26 2021

A320080 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, 1, 0, 1, 3, 6, 2, 0, 1, 4, 15, 28, 4, 0, 1, 5, 28, 114, 172, 14, 0, 1, 6, 45, 296, 1152, 1328, 38, 0, 1, 7, 66, 610, 4168, 14562, 12272, 216, 0, 1, 8, 91, 1092, 11020, 73376, 220842, 132480, 600, 0, 1, 9, 120, 1778, 24084, 248870, 1550048, 3907656, 1633344, 6240, 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,   1,     6,     15,     28,      45,  ...
  0,   2,    28,    114,    296,     610,  ...
  0,   4,   172,   1152,   4168,   11020,  ...
  0,  14,  1328,  14562,  73376,  248870,  ...

Crossrefs

Columns k=0..5 give A000007, A006252, A088501, A335531, A354147, A365604.
Main diagonal gives A317172.
Cf. A048594, A048994, A094416, A320079, A334369.

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} (-1)^(j-1) * (j-1)! * binomial(n,j) * A(n-j,k). - Seiichi Manyama, May 22 2022

A320343 Expansion of e.g.f. 1/sqrt(1 - 2*log(1 + x)).

Original entry on oeis.org

1, 1, 2, 8, 42, 294, 2472, 24828, 286164, 3751428, 54864408, 887989200, 15731200680, 303068103480, 6304498706880, 140890167340560, 3365469544248720, 85585469309951760, 2308349518803845280, 65819488298810181120, 1978202007765686904480, 62505106242073569018720, 2071320752120227622985600
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 22 2019

Keywords

Crossrefs

Cf. A001147, A006252, A048994, A088501, A305404, A346978.

Programs

  • Maple
    seq(n!*coeff(series(1/sqrt(1-2*log(1+x)),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 29 2019
  • Mathematica
    nmax = 22; CoefficientList[Series[1/Sqrt[1 - 2 Log[1 + x]], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] (2 k - 1)!!, {k, 0, n}], {n, 0, 22}]

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*A001147(k).
a(n) ~ n^n / ((exp(1/2) - 1)^(n + 1/2) * exp(n - 1/4)). - Vaclav Kotesovec, Jan 29 2019
a(0) = 1; a(n) = Sum_{k=1..n} (-1)^(k-1) * (2 - k/n) * (k-1)! * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 11 2023

A317172 a(n) = n! * [x^n] 1/(1 - n*log(1 + x)).

Original entry on oeis.org

1, 1, 6, 114, 4168, 248870, 21966768, 2685571560, 434202400896, 89679267601632, 23032451508686400, 7199033431349412576, 2690461258552995849216, 1184680716090974803461072, 606986901206377433194091520, 358023049940533240478842992000, 240858598980174362552808566194176
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 23 2018

Keywords

Links

Crossrefs

Cf. A006252, A088501, A094420, A242817, A317171.
Main diagonal of A320080.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 - n Log[1 + x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[StirlingS1[n, k] n^k k!, {k, n}], {n, 16}]]
  • PARI
    {a(n) = sum(k=0, n, k!*n^k*stirling(n, k, 1))} \\ Seiichi Manyama, Jun 12 2020

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k)*n^k*k!.
a(n) ~ sqrt(2*Pi) * n^(2*n + 1/2) / exp(n + 1/2). - Vaclav Kotesovec, Jul 23 2018

A350719 a(n) = Sum_{k=0..n} k! * 2^k * k^n * Stirling1(n,k).

Original entry on oeis.org

1, 2, 30, 1108, 76372, 8463328, 1375868768, 308440047648, 91189383264864, 34376022491122368, 16093445542120281792, 9160424435706947112576, 6230035512106223752576896, 4989402076922846372194268160, 4647526704475074504983564884992
Offset: 0

Views

Author

Seiichi Manyama, Feb 03 2022

Keywords

Links

Crossrefs

Cf. A320083, A350720.
Cf. A088501, A195005, A350721.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * 2^k * k^n * StirlingS1[n, k], {k, 1, n}]; Array[a, 15, 0] (* Amiram Eldar, Feb 03 2022 *)
  • PARI
    a(n) = sum(k=0, n, k!*2^k*k^n*stirling(n, k, 1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (2*log(1+k*x))^k)))

Formula

E.g.f.: Sum_{k>=0} (2 * log(1 + k*x))^k.

A354237 Expansion of e.g.f. 1 / (1 - log(1 + 2*x) / 2).

Original entry on oeis.org

1, 1, 0, 2, -8, 64, -592, 6768, -90624, 1395840, -24292608, 471453696, -10094066688, 236340378624, -6007053852672, 164713554069504, -4846361933021184, 152300800682754048, -5091189648734748672, 180386551596145508352, -6752521487083688165376
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 06 2022

Keywords

Crossrefs

Cf. A006252, A088500, A088501, A122704, A155585, A227917, A354750, A354751.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[1/(1 - Log[1 + 2 x]/2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS1[n, k] k! 2^(n - k), {k, 0, n}], {n, 0, 20}]
  • PARI
    my(x='x + O('x^20)); Vec(serlaplace(1/(1-log(1+2*x)/2))) \\ Michel Marcus, Jun 06 2022

Formula

a(n) = Sum_{k=0..n} Stirling1(n,k) * k! * 2^(n-k).
a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * (k-1)! * (-2)^(k-1) * a(n-k).
a(n) ~ n! * (-1)^(n+1) * 2^(n+1) / (n * log(n)^2) * (1 - (4 + 2*gamma)/log(n) + (12 + 12*gamma + 3*gamma^2 - Pi^2/2)/log(n)^2 + (2*Pi^2*gamma - 32 + 4*Pi^2 - 24*gamma^2 - 8*zeta(3) - 4*gamma^3 - 48*gamma)/log(n)^3 + (80 - 20*Pi^2*gamma + 40*zeta(3)*gamma - 5*Pi^2*gamma^2 + 160*gamma + 5*gamma^4 + 80*zeta(3) + 40*gamma^3 + Pi^4/12 - 20*Pi^2 + 120*gamma^2)/log(n)^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 06 2022

A354288 Expansion of e.g.f. (1 + x)^(2/(1 - 2 * log(1+x))).

Original entry on oeis.org

1, 2, 10, 72, 664, 7440, 97712, 1468768, 24825184, 465516672, 9582002688, 214642099584, 5195322070656, 135064965744384, 3752151488840448, 110892824334154752, 3473236656134243328, 114893633354895538176, 4002000861023966189568, 146388324613230926979072
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A088819, A354289.
Cf. A000262, A088501, A354286, A354290.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(1+x)^(2/(1-2Log[1+x])),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 13 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(2/(1-2*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, 2^k*k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A088501(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 2^k * A000262(k) * Stirling1(n,k).
a(n) ~ exp(-7/8 + 1/(4*(exp(1/2) - 1)) + sqrt((2*n)/(exp(1/2) - 1))*exp(1/4) - n) * n^(n - 1/4) / (2^(3/4) * (exp(1/2) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

A308878 Expansion of e.g.f. (1 - log(1 + x))/(1 - 2*log(1 + x)).

Original entry on oeis.org

1, 1, 3, 14, 86, 664, 6136, 66240, 816672, 11331552, 174662304, 2961774144, 54785368128, 1097882522112, 23693117756928, 547844658441216, 13511950038494208, 354086653712228352, 9824794572366544896, 287752569360558907392, 8871374335098501292032
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

Comments

Inverse Stirling transform of A002866.

Crossrefs

Cf. A002866, A008275, A011782, A050351, A088501, A306042, A308877.

Programs

  • Mathematica
    nmax = 20; CoefficientList[Series[(1 - Log[1 + x])/(1 - 2 Log[1 + x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[StirlingS1[n, k] 2^(k - 1) k!, {k, 1, n}], {n, 1, 20}]]

Formula

a(0) = 1; a(n) = Sum_{k=1..n} Stirling1(n,k) * 2^(k-1) * k!.
a(n) ~ n! * exp(1/2) / (4 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Jun 29 2019

A335530 Expansion of e.g.f. (1 - 2*log(1 + x))/(1 - 3*log(1 + x)).

Original entry on oeis.org

1, 1, 5, 38, 384, 4854, 73614, 1302552, 26339832, 599220000, 15146634096, 421152109344, 12774687166224, 419781904240464, 14855313525059664, 563252540698636416, 22779973705779470592, 978886224493465845888, 44538419222894143142784
Offset: 0

Views

Author

Seiichi Manyama, Jun 12 2020

Keywords

Crossrefs

Column k=3 of A334369.
Cf. A088501, A335531.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[k! * 3^(k - 1) * StirlingS1[n, k], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Jun 12 2020 *)
With[{nn=20},CoefficientList[Series[(1-2Log[1+x])/(1-3Log[1+x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 12 2021 *)
  • PARI
    {a(n) = if(n==0, 1, sum(k=0, n, k!*3^(k-1)*stirling(n, k, 1)))}
  • PARI
    N=40; x='x+O('x^N); Vec(serlaplace((1-2*log(1+x))/(1-3*log(1+x))))

Formula

a(0)=1 and a(n) = Sum_{k=0..n} k! * 3^(k-1) * Stirling1(n,k) for n > 0.
a(n) ~ n! * exp(1/3) / (9*(exp(1/3)-1)^(n+1)). - Vaclav Kotesovec, Jun 12 2020
Showing 1-9 of 9 results.

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