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 18 results. Next

A346978 Expansion of e.g.f. 1 / sqrt(1 + 2 * log(1 - x)).

Original entry on oeis.org

1, 1, 4, 26, 234, 2694, 37812, 626352, 11962164, 258787812, 6255195168, 167072685240, 4886611129320, 155335056242040, 5332298685827760, 196590247328769120, 7747254471910795920, 324986515253994589200, 14458392906960271354560, 679977065168639138610720
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2021

Keywords

Links

Crossrefs

Cf. A001147, A007840, A088500, A305404, A320343.

Programs

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

Formula

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

A343707 a(n) = 1 + 2 * Sum_{k=0..n-1} binomial(n,k) * (n-k-1)! * a(k).

Original entry on oeis.org

1, 3, 15, 113, 1145, 14539, 221663, 3943281, 80173345, 1833831619, 46606646175, 1302954958689, 39737420405753, 1312901360002283, 46714233470065999, 1780859204826798401, 72416689888874547969, 3128792006916853876291, 143132514626658326870767, 6911638338982428907738641
Offset: 0

Views

Author

Ilya Gutkovskiy, Apr 26 2021

Keywords

Links

Crossrefs

Cf. A088500, A201339, A291979, A343709, A343710.

Programs

  • Mathematica
    a[n_] := a[n] = 1 + 2 Sum[Binomial[n, k] (n - k - 1)! a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 19}]
nmax = 19; CoefficientList[Series[Exp[x]/(1 + 2 Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
  • PARI
    N=20; x='x+O('x^N); Vec(serlaplace(exp(x)/(1+2*log(1-x)))) \\ Seiichi Manyama, Oct 20 2021

Formula

E.g.f.: exp(x) / (1 + 2 * log(1 - x)).
a(n) = Sum_{k=0..n} binomial(n,k) * A088500(k).

A367474 Expansion of e.g.f. 1 / (1 + 2 * log(1 - x))^2.

Original entry on oeis.org

1, 4, 28, 272, 3360, 50256, 881616, 17734944, 402278496, 10155145344, 282329361024, 8570500876032, 282047266728192, 10001430040080384, 380152962804068352, 15418451851593596928, 664633482628021493760, 30342827915683778027520
Offset: 0

Views

Author

Seiichi Manyama, Nov 19 2023

Keywords

Crossrefs

Cf. A088500, A367475.
Cf. A052801, A367470.

Programs

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

Formula

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

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

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).

A317171 a(n) = n! * [x^n] 1/(1 + n*log(1 - x)).

Original entry on oeis.org

1, 1, 10, 222, 8824, 553870, 50545008, 6328330344, 1041597412224, 218138133235680, 56650689388344000, 17868469522986145536, 6728682216722958185472, 2981868816113406609186576, 1536217706761623823662025728, 910442461680276910819097616000, 615053979239579281793375485526016
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 23 2018

Keywords

Crossrefs

Cf. A007840, A088500, A094420, A242817, A317172.
Main diagonal of A320079.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(1 + n Log[1 - x]), {x, 0, n}], {n, 0, 16}]
Join[{1}, Table[Sum[Abs[StirlingS1[n, k]] n^k k!, {k, n}], {n, 16}]]

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

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

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

A354286 Expansion of e.g.f. 1/(1 - x)^(2/(1 + 2 * log(1-x))).

Original entry on oeis.org

1, 2, 14, 144, 1936, 32000, 625952, 14117152, 360175584, 10246079616, 321313928448, 11006050602624, 408662128569984, 16344011453662464, 700254206319007488, 31990601456727585792, 1551985176120589820928, 79669906174753878177792
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A088815, A354287.
Cf. A000262, A088500, A354288, A354290.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(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!*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} A088500(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 2^k * A000262(k) * |Stirling1(n,k)|.
a(n) ~ n^(n - 1/4) / (2^(3/4) * (exp(1/2) - 1)^(n + 1/4) * exp(3/4 - 1/(4*(exp(1/2) - 1)) - sqrt(2*n/(exp(1/2) - 1)) + n/2)). - Vaclav Kotesovec, May 23 2022

A368285 Expansion of e.g.f. exp(2*x) / (1 + 2*log(1 - x)).

Original entry on oeis.org

1, 4, 22, 168, 1700, 21560, 328576, 5844608, 118827264, 2717955776, 69076424384, 1931128212992, 58895387322240, 1945869352171264, 69235812945551872, 2639436090012161024, 107329778640349652992, 4637225944423696109568, 212138681191492565180416
Offset: 0

Views

Author

Seiichi Manyama, Dec 19 2023

Keywords

Crossrefs

Cf. A088500, A343707, A368286, A368287.

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).
a(n) ~ n! * exp(n/2 + 2 - 2*exp(-1/2)) / (2 * (exp(1/2) - 1)^(n+1)). - Vaclav Kotesovec, Dec 29 2023
Showing 1-10 of 18 results. Next

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

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