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.

A352309 Expansion of e.g.f. 1/(exp(x) - x^2/2).

Original entry on oeis.org

1, -1, 2, -7, 31, -171, 1141, -8863, 78653, -785557, 8716861, -106395741, 1416724915, -20436548575, 317477947151, -5284248213091, 93816998697721, -1769737117839849, 35347571931577609, -745232024035027225, 16538641134235561631, -385387334950748244451
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Cf. A089148, A352310, A352311.
Cf. A352302, A352306.

Programs

  • Mathematica
    m = 21; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^2/2), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^2/2)))
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m))*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 2);

Formula

a(n) = binomial(n,2) * a(n-2) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 1.
a(n) ~ n! * (-1)^n / ((1 + LambertW(1/sqrt(2))) * (2*LambertW(1/sqrt(2)))^(n+2)). - Vaclav Kotesovec, Mar 12 2022
a(n) = n! * Sum_{k=0..floor(n/2)} (-k-1)^(n-2*k)/(2^k*(n-2*k)!). - Seiichi Manyama, Aug 21 2024

A352311 Expansion of e.g.f. 1/(exp(x) - x^4/24).

Original entry on oeis.org

1, -1, 1, -1, 2, -11, 61, -281, 1191, -5923, 41791, -354091, 2968021, -24059751, 204718515, -1996937671, 22125450621, -258434553861, 3056858429581, -37181421375349, 482010195953821, -6741275765687821, 99663246605243861, -1521712424934601901
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Cf. A089148, A352309, A352310.
Cf. A352304.

Programs

  • Mathematica
    m = 23; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^4/24), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^4/24)))
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m))*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 4);

Formula

a(n) = binomial(n,4) * a(n-4) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 3.
a(n) ~ n! * 3*(-1)^n / ((1 + LambertW(3^(1/4) / 2^(5/4))) * 2^(2*n + 7) * LambertW(3^(1/4) / 2^(5/4))^(n+4)). - Vaclav Kotesovec, Mar 12 2022
a(n) = n! * Sum_{k=0..floor(n/4)} (-k-1)^(n-4*k)/(24^k*(n-4*k)!). - Seiichi Manyama, Aug 21 2024

A352310 Expansion of e.g.f. 1/(exp(x) - x^3/6).

Original entry on oeis.org

1, -1, 1, 0, -7, 39, -139, 139, 3249, -38305, 257641, -724681, -9925519, 208718223, -2209932451, 11619569779, 98841199521, -3691083087521, 56488651405393, -466578080641297, -1989509977776479, 159427986446212959, -3372599255892634459, 39809520784433784075
Offset: 0

Views

Author

Seiichi Manyama, Mar 11 2022

Keywords

Crossrefs

Cf. A089148, A352309, A352311.
Cf. A352303.

Programs

  • Mathematica
    m = 23; Range[0, m]! * CoefficientList[Series[1/(Exp[x] - x^3/6), {x, 0, m}], x] (* Amiram Eldar, Mar 12 2022 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(exp(x)-x^3/6)))
  • PARI
    b(n, m) = if(n==0, 1, sum(k=1, n, (-1+(k==m))*binomial(n, k)*b(n-k, m)));
a(n) = b(n, 3);

Formula

a(n) = binomial(n,3) * a(n-3) - Sum_{k=1..n} binomial(n,k) * a(n-k) for n > 2.
a(n) = n! * Sum_{k=0..floor(n/3)} (-k-1)^(n-3*k)/(6^k*(n-3*k)!). - Seiichi Manyama, Aug 21 2024

A375608 Expansion of e.g.f. 1 / (exp(x^2) - x).

Original entry on oeis.org

1, 1, 0, -6, -36, -120, 240, 8400, 82320, 362880, -3507840, -103783680, -1268688960, -4843238400, 175429013760, 5052189542400, 68016191443200, 55329155481600, -23284682272051200, -668640423164313600, -9013925405784499200, 57340797108269875200
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Cf. A089148, A375609.
Cf. A375394, A375604.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(exp(x^2)-x)))
  • PARI
    a(n) = n!*sum(k=0, n\2, (-n+2*k-1)^k/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (-n+2*k-1)^k/k!.

A338448 E.g.f.: 1 / (1 - x - log(1 - x)).

Original entry on oeis.org

1, 0, -1, -2, 0, 16, 50, -132, -2184, -9984, 6912, 341760, 38544, -47086272, -702019344, -6076389984, -43980940800, -656377887744, -16782743357568, -368775477229824, -6770025717901056, -118247220867640320, -2271088046291742720, -50203882870716579840
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 28 2020

Keywords

Crossrefs

Cf. A006252, A089148, A226226.

Programs

  • Mathematica
    nmax = 23; CoefficientList[Series[1/(1 - x - Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 2, n}]; Table[a[n], {n, 0, 23}]
  • PARI
    my(x='x + O('x^30)); Vec(serlaplace(1/(1 - x - log(1 - x)))) \\ Michel Marcus, Oct 29 2020

Formula

a(0) = 1; a(n) = -Sum_{k=2..n} binomial(n,k) * (k-1)! * a(n-k).
a(n) ~ -n! / (n * log(n)^2) * (1 - 2*gamma/log(n) + (3*gamma^2 - Pi^2/2)/log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Oct 29 2020

A375609 Expansion of e.g.f. 1 / (exp(x^3) - x).

Original entry on oeis.org

1, 1, 2, 0, -24, -240, -1800, -10080, -20160, 665280, 15120000, 219542400, 2335132800, 11416204800, -272432160000, -11126129414400, -252293974732800, -4099297608806400, -36217872365875200, 593695670606438400, 41572213064718336000, 1335024565711828992000
Offset: 0

Views

Author

Seiichi Manyama, Aug 21 2024

Keywords

Crossrefs

Cf. A089148, A375608.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(exp(x^3)-x)))
  • PARI
    a(n) = n!*sum(k=0, n\3, (-n+3*k-1)^k/k!);

Formula

a(n) = n! * Sum_{k=0..floor(n/3)} (-n+3*k-1)^k/k!.

A144452 Antidiagonal expansion of the polynomials: f(x,n) = 1/(exp(t) - Sum_{i=1..n} t^i/i!).

Original entry on oeis.org

1, 1, 0, 1, 0, -3, 1, 0, 0, -4, 1, 0, 0, -4, 25, 1, 0, 0, 0, -5, 114, 1, 0, 0, 0, -5, -6, -287, 1, 0, 0, 0, 0, -6, 133, -4152, 1, 0, 0, 0, 0, -6, -7, 552, -1647, 1, 0, 0, 0, 0, 0, -7, -8, 1629, 192230, 1, 0, 0, 0, 0, 0, -7, -8, 621, -12610, 807961, 1, 0, 0, 0, 0, 0, 0, -8, -9, 2510, -128579, -10164804, 1, 0, 0, 0, 0, 0, 0, -8, -9, -10, 7381
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Oct 06 2008

Keywords

Comments

Row sums are:
{1, 2, 3, 20, 125, 804, 4501, 36896, 362673, 3831560, 40591001, 467518248, 6106124713, 87661533764, 1323052370025}.
Triangle sequence rows last terms are:
Table[n!*a[[1]][[n]], {n, 1, 15}]
{1, 0, -3, -4, 25,114, -287, -4152, -1647, 192230, 807961, -10164804, -111209111, 454840554, 14657978385}

Examples

			{1},
{1, 0},
{1, 0, -3},
{1, 0, 0, -4},
{1, 0, 0, -4, 25},
{1, 0, 0, 0, -5, 114},
{1, 0, 0, 0, -5, -6, -287},
{1, 0, 0, 0, 0, -6, 133, -4152},
{1, 0, 0, 0, 0, -6, -7, 552, -1647},
{1, 0, 0, 0, 0, 0, -7, -8,1629, 192230},
{1, 0, 0, 0, 0, 0, -7, -8, 621, -12610, 807961},
{1, 0, 0, 0, 0, 0, 0, -8, -9, 2510, -128579, -10164804},
{1, 0, 0, 0, 0, 0, 0, -8, -9, -10, 7381, -725484, -111209111},
{1, 0, 0, 0, 0,0, 0, 0, -9, -10, 2761, 18996, 1522651, 454840554},
{1, 0, 0, 0, 0, 0, 0,0, -9, -10, -11, 11076, -404989, 54082014, 14657978385}

Crossrefs

Cf. A089148.

Programs

  • Mathematica
    Clear[f, b, a, g, h, n, t]; f[t_, n_] = 1/(Exp[t] - Sum[t^i/i!, {i, 1, n}]); a = Table[Table[SeriesCoefficient[Series[f[t, m], {t, 0, 30}], n], {n, 0, 30}], {m, 1, 31}]; b = Table[Table[m!*a[[n - m + 1]][[m]], {m, 1, n }], {n, 1, 15}]; Flatten[b]

Formula

f(x,n) = 1/(exp(t) - Sum_{i=1..n} t^i/i!); t(n,m) = Expansion(f(x,n)); t_out(n,m) = m!*t(n-m+1,m).

A336969 a(n) = n! * [x^n] 1 / (exp(n*x) - x).

Original entry on oeis.org

1, 0, -2, 33, -424, 495, 342864, -22382913, 915074432, -913039857, -5455432211200, 812138028148623, -75257247474017280, 1984517460320303415, 1155562494647499610112, -361521639388178369672625, 67461150715150454861692928, -6658374003334822571921759457
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2020

Keywords

Crossrefs

Cf. A089148, A134095, A302398, A336949, A336958.

Programs

  • Mathematica
    Table[n! SeriesCoefficient[1/(Exp[n x] - x), {x, 0, n}], {n, 0, 17}]
Join[{1}, Table[n! Sum[(-n (n - k + 1))^k/k!, {k, 0, n}], {n, 1, 17}]]

Formula

a(n) = n! * Sum_{k=0..n} (-n * (n-k+1))^k / k!.
Previous Showing 11-18 of 18 results.

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