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 1-7 of 7 results.

A367470 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^2.

Original entry on oeis.org

1, 4, 28, 268, 3244, 47404, 810988, 15891628, 350851564, 8615761324, 232911898348, 6872755977388, 219799913877484, 7572909749244844, 279630706025296108, 11016315458773541548, 461211305514352065004, 20448268640012928321964
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A004123, A367471.
Cf. A005649, A367472.
Cf. A367474.

Programs

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

Formula

a(n) = Sum_{k=0..n} 2^k * (k+1)! * Stirling2(n,k).
a(0) = 1; a(n) = 2*Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * log(3/2)^(n+2) * exp(n)). - Vaclav Kotesovec, May 20 2025

A367475 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^3.

Original entry on oeis.org

1, 6, 54, 636, 9204, 157584, 3111312, 69533472, 1734229344, 47733263232, 1436801816448, 46942939272960, 1654215709835520, 62533593070755840, 2524077593084160000, 108339176213529384960, 4927173048408858531840, 236673892535088351744000
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A088500, A367474.
Cf. A354122, A367471.

Programs

  • Maple
    A367475 := proc(n)
    option remember ;
    if n =0 then
        1;
    else
        2*add((2*k/n + 1) * (k-1)! * binomial(n,k) * procname(n-k),k=1..n) ;
    end if;
end proc:
seq(A367475(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    a(n) = sum(k=0, n, 2^k*(k+2)!*abs(stirling(n, k, 1)))/2;

Formula

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

A375945 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^(3/2).

Original entry on oeis.org

1, 3, 18, 156, 1758, 24342, 399480, 7577700, 163090500, 3926104860, 104520733560, 3048811591680, 96695722690200, 3312942954681240, 121938065727180480, 4798400761979259120, 201030443703421854480, 8933622147642363338160, 419725992843354254228640
Offset: 0

Views

Author

Seiichi Manyama, Sep 03 2024

Keywords

Crossrefs

Cf. A052801, A375946.
Cf. A088500, A367474, A367475, A375953.
Cf. A001147.

Programs

  • Mathematica
    nmax=18; CoefficientList[Series[1 / (1 + 2 * Log[1 - x])^(3/2),{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+1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} A001147(k+1) * |Stirling1(n,k)|.
a(n) ~ n^(n+1) / (exp(n/2) * (exp(1/2) - 1)^(n + 3/2)). - Vaclav Kotesovec, Sep 06 2024

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)|.

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)|.

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

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.
Showing 1-7 of 7 results.

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

Content is available under The OEIS End-User License Agreement.