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 11-18 of 18 results.

A368286 Expansion of e.g.f. exp(-x) / (1 + 2*log(1 - x)).

Original entry on oeis.org

1, 1, 7, 51, 521, 6617, 100903, 1795091, 36497601, 834825089, 21217022903, 593152248323, 18089914384425, 597680325734905, 21266014041519799, 810711731123810051, 32966705053762073665, 1424339658492670445121, 65159114638457033834791
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A088500, A343707, A368285, A368287.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-1)^i+2*sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = (-1)^n + 2 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).

A375953 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^(5/2).

Original entry on oeis.org

1, 5, 40, 430, 5770, 92590, 1726940, 36682200, 873793620, 23061929940, 667868085360, 21052931727240, 717531427466280, 26289935772108120, 1030422613932910800, 43018144091244322560, 1905711682795871222160, 89284805444478025826640
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Cf. A088500, A367474, A367475, A375945.
Cf. A001147.

Programs

  • Mathematica
    nmax=17; CoefficientList[Series[1 / (1 + 2 * Log[1 - x])^(5/2),{x,0,nmax}],x]*Range[0,nmax]! (* Stefano Spezia, Sep 03 2024 *)
  • PARI
    a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k+2)*abs(stirling(n, k, 1)))/3;

Formula

a(n) = (1/3) * Sum_{k=0..n} A001147(k+2) * |Stirling1(n,k)|.

A355231 E.g.f. A(x) satisfies A'(x) = 1 - 2 * log(1-x) * A(x).

Original entry on oeis.org

0, 1, 0, 4, 6, 48, 200, 1364, 9016, 71088, 607920, 5772528, 59790720, 673839456, 8210152704, 107668087104, 1513106471040, 22700196933120, 362277092798208, 6130771723664640, 109694104262443008, 2069581743476587008, 41071931895114372096, 855436794313229319168
Offset: 0

Views

Author

Seiichi Manyama, Jun 25 2022

Keywords

Crossrefs

Cf. A088500, A355205, A355229, A355230.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[(1-x)^(2 - 2*x)/E^(2 - 2*x) * Integrate[E^(2 - 2*x) / (1-x)^(2 - 2*x), x], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Jun 25 2022 *)
  • PARI
    a_vector(n) = my(v=vector(n)); v[1]=1; for(i=1, n-1, v[i+1]=2*sum(j=1, i-1, (j-1)!*binomial(i, j)*v[i-j])); concat(0, v);

Formula

a(0) = 0, a(1) = 1; a(n+1) = 2 * Sum_{k=1..n-1} (k-1)! * binomial(n,k) * a(n-k).
E.g.f.: (1-x)^(2 - 2*x) / exp(2 - 2*x) * Integral(exp(2 - 2*x) / (1-x)^(2 - 2*x) dx). - Vaclav Kotesovec, Jun 25 2022

A368287 Expansion of e.g.f. exp(-2*x) / (1 + 2*log(1 - x)).

Original entry on oeis.org

1, 0, 6, 32, 356, 4456, 68096, 1211136, 24625408, 563266240, 14315378880, 400206928128, 12205482237824, 403262088466688, 14348434923733504, 546996936260529152, 22243031618999642112, 961019064912965103616, 43963636798214215278592
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A088500, A343707, A368285, A368286.

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); for(i=0, n, v[i+1]=(-2)^i+2*sum(j=1, i, (j-1)!*binomial(i, j)*v[i-j+1])); v;

Formula

a(n) = (-2)^n + 2 * Sum_{k=1..n} (k-1)! * binomial(n,k) * a(n-k).

A375987 Expansion of e.g.f. (1 + 2 * log(1 - x))^(3/2).

Original entry on oeis.org

1, -3, 0, 6, 42, 318, 2892, 31944, 424596, 6682740, 122318928, 2559121128, 60275236392, 1577894836248, 45427570253712, 1425885338250432, 48443767097018256, 1770703320887526096, 69273368628184075392, 2887794188011931364576, 127778992241790634125984
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Cf. A088500, A367474, A375945, A375953, A375990.

Programs

  • Maple
    A375987 := proc(n)
    add(mul(2*j-3,j=0..k-1)*abs(stirling1(n,k)),k=0..n) ;
end proc:
seq(A375987(n),n=0..30) ; # R. J. Mathar, Sep 06 2024
  • PARI
    a(n) = sum(k=0, n, prod(j=0, k-1, 2*j-3)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (2*j-3)) * |Stirling1(n,k)|.

A308877 Expansion of e.g.f. (1 + log(1 - x))/(1 + 2*log(1 - x)).

Original entry on oeis.org

1, 1, 5, 38, 386, 4904, 74776, 1330272, 27046848, 618653280, 15723024864, 439559609664, 13405656582336, 442915145716224, 15759326934391296, 600783539885546496, 24430204949876794368, 1055516761826050203648, 48286612866726631489536, 2331682676308057000255488
Offset: 0

Views

Author

Ilya Gutkovskiy, Jun 29 2019

Keywords

Crossrefs

Cf. A001710, A002866, A008275, A011782, A050351, A088500, A320349, A308878.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[(1 + Log[1 - x])/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] 2^(k - 1) k!, {k, 1, n}], {n, 1, 19}]]

Formula

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

A371297 E.g.f. satisfies A(x) = 1/(1 + 2*log(1 - x*A(x)^2)).

Original entry on oeis.org

1, 2, 26, 676, 26852, 1443888, 98183024, 8083614880, 781958648448, 86940057459840, 10925288128027968, 1531414930604605440, 236905910564035082112, 40093453025252047368192, 7368774639911257328778240, 1461607086204159742139338752, 311206233406111454756938844160
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2024

Keywords

Crossrefs

Cf. A088500, A370941.
Cf. A367138.

Programs

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

Formula

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

A375990 Expansion of e.g.f. (1 + 2 * log(1 - x))^2.

Original entry on oeis.org

1, -4, 4, 16, 64, 304, 1712, 11232, 84384, 715392, 6761088, 70513920, 804683520, 9975536640, 133513989120, 1919012014080, 29482606540800, 482183099596800, 8364495012249600, 153406409645260800, 2965940772905779200, 60291976261386240000
Offset: 0

Views

Author

Seiichi Manyama, Sep 05 2024

Keywords

Crossrefs

Cf. A088500, A367474, A375945, A375953, A375987.
Cf. A182541.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[(1+2Log[1-x])^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 05 2025 *)
  • PARI
    a(n) = if(n==0, 1, -4*(n-1)!+8*abs(stirling(n, 2, 1)));

Formula

a(n) = -4 * (n-1)! + 8 * |Stirling1(n,2)| for n > 0.
Previous Showing 11-18 of 18 results.

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