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.

A292953 E.g.f.: exp(-1/2! * x^2 * exp(x)).

Original entry on oeis.org

1, 0, -1, -3, -3, 20, 150, 504, -343, -18180, -140220, -500500, 2032899, 50210082, 441768236, 1740141480, -13025325615, -330558552376, -3452606080848, -16648495695792, 136964192085395, 4315989335784630, 55121200672923924, 352945156766431592
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=2 of A293015.
Cf. A292952 (k=1), this sequence (k=2), A292954 (k=3), A292955 (k=4).

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-1/2!*x^2*exp(x))))

A292954 E.g.f.: exp(-1/3! * x^3 * exp(x)).

Original entry on oeis.org

1, 0, 0, -1, -4, -10, -10, 105, 1064, 6356, 25080, 9075, -1056660, -13219206, -106106364, -548948855, 139658960, 48411569800, 761039099824, 7815284148711, 52216924707660, 9385130453790, -6650556642220260, -132749143322588331, -1713641693856894824
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=3 of A293015.
Cf. A292952 (k=1), A292953 (k=2), this sequence (k=3), A292955 (k=4).
Cf. A292910.

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-1/3!*x^3*exp(x))))

Formula

a(n) = (-1)^n * A292910(n).

A292955 E.g.f.: exp(-1/4! * x^4 * exp(x)).

Original entry on oeis.org

1, 0, 0, 0, -1, -5, -15, -35, -35, 504, 6090, 45870, 270930, 1215500, 1995994, -42118440, -733409495, -8069463780, -70153266240, -468024155016, -1498366231020, 21132982355940, 568009017066260, 8607952077741940, 101448276642079059, 937291639168833850
Offset: 0

Views

Author

Seiichi Manyama, Sep 27 2017

Keywords

Links

Crossrefs

Column k=4 of A293015.
Cf. A292952 (k=1), A292953 (k=2), A292954 (k=3), this sequence (k=4).

Programs

  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(-1/4!*x^4*exp(x))))

A293015 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = - Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, 0, -1, 2, 1, 1, 0, 0, -3, 9, -2, 1, 0, 0, -1, -3, 4, -9, 1, 0, 0, 0, -4, 20, -95, -9, 1, 0, 0, 0, -1, -10, 150, -414, 50, 1, 0, 0, 0, 0, -5, -10, 504, 49, 267, 1, 0, 0, 0, 0, -1, -15, 105, -343, 10088, 413, 1, 0, 0, 0, 0, 0, -6
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2017

Keywords

Examples

			Square array begins:
    1,  1,  1,  1,  1, ...
   -1, -1,  0,  0,  0, ...
    0, -1, -1,  0,  0, ...
    1,  2, -3, -1,  0, ...
    1,  9, -3, -4, -1, ...

Links

Crossrefs

Columns k=0-4 give: A000587, A292952, A292953, A292954, A292955.
Rows n=0 gives A000012.
Cf. A145460.

A293019 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where A(0,k) = 1 and A(n,k) = - k! * Sum_{i=0..n-1} binomial(n-1,i) * binomial(i+1,k) * A(n-1-i,k) for n > 0.

Original entry on oeis.org

1, 1, -1, 1, -1, 0, 1, 0, -1, 1, 1, 0, -2, 2, 1, 1, 0, 0, -6, 9, -2, 1, 0, 0, -6, 0, 4, -9, 1, 0, 0, 0, -24, 100, -95, -9, 1, 0, 0, 0, -24, -60, 570, -414, 50, 1, 0, 0, 0, 0, -120, 240, 798, 49, 267, 1, 0, 0, 0, 0, -120, -360, 4830, -15176, 10088, 413, 1, 0, 0
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2017

Keywords

Examples

			Square array begins:
    1,  1,  1,   1,   1, ...
   -1, -1,  0,   0,   0, ...
    0, -1, -2,   0,   0, ...
    1,  2, -6,  -6,   0, ...
    1,  9,  0, -24, -24, ...

Links

Crossrefs

Columns k=0-4 give: A000587, A292952, A293016, A293017, A293018.
Rows n=0 gives A000012.
Cf. A292978, A293015.

A356819 Expansion of e.g.f. exp(-x * exp(2*x)).

Original entry on oeis.org

1, -1, -3, -1, 41, 239, 229, -8401, -87151, -324577, 3238541, 70271519, 601086265, 142860431, -81504662539, -1393683935281, -10777424809951, 63537986981183, 3552608426329117, 60283510555017023, 441644419610814281, -6191820436867600081
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Crossrefs

Cf. A216689, A292952, A356812, A356820.

Programs

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

Formula

G.f.: Sum_{k>=0} (-x)^k / (1 - 2*k*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (2*k)^(n-k) * binomial(n,k).

A356820 Expansion of e.g.f. exp(-x * exp(3*x)).

Original entry on oeis.org

1, -1, -5, -10, 73, 1004, 5473, -15562, -746447, -9174088, -41916959, 823985546, 24629093641, 335144105828, 1248594602305, -67564407472426, -2160461588461343, -34957074099518608, -154556217713939903, 10500560586914149250, 409146670525578079801
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Crossrefs

Cf. A292952, A356813, A356819.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[Exp[-x Exp[3x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Mar 13 2025 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*exp(3*x))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-3*k*x)^(k+1)))
  • PARI
    a(n) = sum(k=0, n, (-1)^k*(3*k)^(n-k)*binomial(n, k));

Formula

G.f.: Sum_{k>=0} (-x)^k / (1 - 3*k*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (3*k)^(n-k) * binomial(n,k).

A328488 Expansion of e.g.f. 1 / (2 - exp(x * exp(x))).

Original entry on oeis.org

1, 1, 5, 34, 307, 3456, 46659, 734882, 13227995, 267871036, 6027206803, 149176155030, 4027831914099, 117816299188472, 3711283196035523, 125258162280991858, 4509378597919760779, 172486973301491042964, 6985853719202139488211, 298650859698906574479278
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 16 2019

Keywords

Crossrefs

Cf. A000248, A000670, A003725, A007550, A083355, A292952.

Programs

  • Mathematica
    nmax = 19; CoefficientList[Series[1/(2 - Exp[x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!

Formula

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n,k) * A000248(k) * a(n-k).
a(n) ~ n! / (2*log(2) * (1 + LambertW(log(2))) * LambertW(log(2))^n). - Vaclav Kotesovec, Oct 17 2019

A336610 Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(-sqrt(x) * BesselI(1,2*sqrt(x))).

Original entry on oeis.org

1, -1, 0, 9, -4, -625, -906, 145187, 1350040, -71822385, -2093778910, 49843036199, 4422338360340, 7491520000835, -11939082153832302, -455740256735697165, 33146485198521406064, 4039886119274766333343, 2019781328116371668154
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 28 2020

Keywords

Crossrefs

Cf. A003725, A292952, A302397, A336209, A336227.

Programs

  • Mathematica
    nmax = 18; CoefficientList[Series[Exp[-Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
a[0] = 1; a[n_] := a[n] = -n Sum[Binomial[n - 1, k]^2 a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = -n * Sum_{k=0..n-1} binomial(n-1,k)^2 * a(k).
Showing 1-9 of 9 results.

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