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.

A346738 Expansion of e.g.f.: exp(exp(x) - 3*x - 1).

Original entry on oeis.org

1, -2, 5, -13, 36, -101, 293, -848, 2523, -7365, 22402, -64395, 205285, -541802, 2057617, -3403993, 28685420, 43885023, 824532745, 4878097904, 44263112047, 357891860463, 3169228222338, 28506399763969, 266822555964441, 2573194635922990, 25606751525353741
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 31 2021

Keywords

Links

Crossrefs

Cf. A000110, A000296, A005494, A126617, A193683, A290219, A346739, A346740.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50);
Coefficients(R!(Laplace( Exp(Exp(x)-3*x-1) ))) // G. C. Greubel, Jun 12 2024
  • Mathematica
    nmax = 26; CoefficientList[Series[Exp[Exp[x] - 3 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k], {k, 0, n}], {n, 0, 26}]
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 26}]
  • SageMath
    [factorial(n)*( exp(exp(x)-3*x-1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

Formula

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

A290219 a(n) = n! * [x^n] exp(exp(x) - n*x - 1).

Original entry on oeis.org

1, 0, 2, -13, 127, -1573, 23711, -421356, 8626668, -199971255, 5177291275, -148078588667, 4636966634653, -157786054331852, 5797411243015250, -228749440644895405, 9646951350227609155, -433035586385769361001, 20614401475233006857035, -1037331650810058231498688
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 06 2017

Keywords

Comments

The n-th term of the n-th inverse binomial transform of A000110.

Links

Crossrefs

Cf. A000110, A000296, A126617, A134980, A346738, A346739, A346740.
Main diagonal of A361781.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50);
A290219:= func< n | Coefficient(R!(Laplace( Exp(Exp(x)-n*x-1) )), n) >;
[A290219(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024
  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1,
      k*b(n-1, k)+ b(n-1, k+1))
    end:
a:= n-> b(n, -n):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021
  • Mathematica
    Table[n! SeriesCoefficient[Exp[Exp[x] - n x - 1], {x, 0, n}], {n, 0, 19}]
Join[{1}, Table[Sum[(-n)^(n - k) Binomial[n, k] BellB[k] , {k, 0, n}], {n, 1, 19}]]
  • SageMath
    [factorial(n)*( exp(exp(x) -n*x -1) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

Formula

a(n) ~ (-1)^n * exp(exp(-1) - 1) * n^n. - Vaclav Kotesovec, Aug 04 2021

A193684 Alternating row sums of Sheffer triangle A143496 (4-restricted Stirling2 numbers).

Original entry on oeis.org

1, 3, 8, 17, 17, -78, -585, -2021, -1710, 29395, 231413, 856264, -346979, -30019585, -232782792, -834712259, 2313820717, 59793779314, 469729578123, 1597321309383, -9914171906614, -206169178856073, -1697255630380351, -5677886943413120, 55801423903125353
Offset: 0

Views

Author

Wolfdieter Lang, Oct 06 2011

Keywords

Comments

In order to have A143496 as a lower triangular Sheffer matrix one uses row and column offsets 0 (not 4).

Examples

			With offset [0,0] row n=3 of A143496 is [64,61,15,1], hence a(3)=64-61+15-1=17.

Links

Crossrefs

Cf. A143496, A193683 (3-restricted Stirling2 case), A196835, A293037, A346739.

Programs

  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+4*x+1))) \\ Michel Marcus, Aug 02 2021

Formula

E.g.f.: exp(-exp(x)+4*x+1).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 4)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 4 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

A346740 Expansion of e.g.f.: exp(exp(x) - 5*x - 1).

Original entry on oeis.org

1, -4, 17, -75, 340, -1573, 7393, -35178, 169035, -818603, 3989250, -19538555, 96084397, -474052868, 2344993157, -11624422855, 57722000172, -287012948441, 1428705217949, -7118044107698, 35489117143047, -177036294035559, 883588566571138, -4411213326568599, 22032317835916969
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 31 2021

Keywords

Links

Crossrefs

Cf. A000110, A000296, A126617, A196834, A196835, A290219, A346738, A346739.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!(Laplace( Exp(Exp(x) -5*x -1) ))) // G. C. Greubel, Jun 12 2024
  • Mathematica
    nmax = 24; CoefficientList[Series[Exp[Exp[x] - 5 x - 1], {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] (-5)^(n - k) BellB[k], {k, 0, n}], {n, 0, 24}]
a[0] = 1; a[n_] := a[n] = -5 a[n - 1] + Sum[Binomial[n - 1, k] a[k], {k, 0, n - 1}]; Table[a[n], {n, 0, 24}]
  • SageMath
    [factorial(n)*( exp(exp(x) -5*x -1) ).series(x, n+1).list()[n] for n in (0..30)] # G. C. Greubel, Jun 12 2024

Formula

G.f. A(x) satisfies: A(x) = (1 - x + x * A(x/(1 - x))) / ((1 - x) * (1 + 5*x)).
a(n) = Sum_{k=0..n} binomial(n,k) * (-5)^(n-k) * Bell(k).
a(n) = exp(-1) * Sum_{k>=0} (k - 5)^n / k!.
a(0) = 1; a(n) = -5 * a(n-1) + Sum_{k=0..n-1} binomial(n-1,k) * a(k).

A367891 Expansion of e.g.f. exp(4*(exp(x) - 1 - x)).

Original entry on oeis.org

1, 0, 4, 4, 52, 164, 1364, 7620, 60148, 449252, 3831700, 33811716, 320082228, 3178774564, 33234163668, 363535920196, 4153091085172, 49406896240996, 610777358429204, 7830140410294148, 103914148870277556, 1425254885630973604, 20173671034640405588
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

Crossrefs

Cf. A000296, A078944, A194689, A346739, A367890.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Exp[4 (Exp[x] - 1 - x)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = 4 Sum[Binomial[n - 1, k] a[n - k - 1], {k, 1, n - 1}]; Table[a[n], {n, 0, 22}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, 4], {k, 0, n}], {n, 0, 22}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(4*(exp(x) - 1 - x)))) \\ Michel Marcus, Dec 04 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 4 * x * ( A(x) - A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-4)^n / k!.
a(0) = 1; a(n) = 4 * Sum_{k=1..n-1} binomial(n-1,k) * a(n-k-1).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A078944(k).

A361781 A(n,k) is the n-th term of the k-th inverse binomial transform of the Bell numbers (A000110); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, -1, 1, 5, 1, -2, 2, 1, 15, 1, -3, 5, -3, 4, 52, 1, -4, 10, -13, 7, 11, 203, 1, -5, 17, -35, 36, -10, 41, 877, 1, -6, 26, -75, 127, -101, 31, 162, 4140, 1, -7, 37, -139, 340, -472, 293, -21, 715, 21147, 1, -8, 50, -233, 759, -1573, 1787, -848, 204, 3425, 115975
Offset: 0

Views

Author

Alois P. Heinz, Mar 23 2023

Keywords

Examples

			Square array A(n,k) begins:
    1,   1,   1,    1,     1,      1,       1,       1, ...
    1,   0,  -1,   -2,    -3,     -4,      -5,      -6, ...
    2,   1,   2,    5,    10,     17,      26,      37, ...
    5,   1,  -3,  -13,   -35,    -75,    -139,    -233, ...
   15,   4,   7,   36,   127,    340,     759,    1492, ...
   52,  11, -10, -101,  -472,  -1573,   -4214,   -9685, ...
  203,  41,  31,  293,  1787,   7393,   23711,   63581, ...
  877, 162, -21, -848, -6855, -35178, -134873, -421356, ...

Links

Crossrefs

Columns k=0-5 give: A000110, A000296, A126617, A346738, A346739, A346740.
Rows n=0-2 give: A000012, A024000, A160457.
Main diagonal gives A290219.
Antidiagonal sums give A361380.
Cf. A108087.

Programs

  • Magma
    T:= func< n,k | (&+[(-k)^j*Binomial(n,j)*Bell(n-j): j in [0..n]]) >;
A361781:= func< n,k | T(k, n-k) >;
[A361781(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 12 2024
  • Maple
    A:= proc(n, k) option remember; uses combinat;
      add(binomial(n, j)*(-k)^j*bell(n-j), j=0..n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
# second Maple program:
b:= proc(n, m) option remember;
     `if`(n=0, 1, b(n-1, m+1)+m*b(n-1, m))
    end:
A:= (n, k)-> b(n, -k):
seq(seq(A(n, d-n), n=0..d), d=0..10);
  • Mathematica
    T[n_, k_]:= T[n, k]= If[k==0, BellB[n], Sum[(-k)^j*Binomial[n,j]*BellB[n-j], {j,0,n}]];
A361781[n_, k_]= T[k, n-k];
Table[A361781[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 12 2024 *)
  • SageMath
    def T(n,k): return sum( (-k)^j*binomial(n,j)*bell_number(n-j) for j in range(n+1))
def A361781(n, k): return T(k, n-k)
flatten([[A361781(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jun 12 2024

Formula

E.g.f. of column k: exp(exp(x) - k*x - 1).
A(n,k) = Sum_{j=0..n} (-k)^j*binomial(n,j)*Bell(n-j).

A367819 Expansion of e.g.f. exp(1 - 4*x - exp(x)).

Original entry on oeis.org

1, -5, 24, -111, 497, -2166, 9239, -38765, 160658, -659773, 2691205, -10922544, 44166173, -178098121, 716703848, -2879774019, 11558005677, -46348854134, 185746261419, -744036460097, 2979305960426, -11926715433881, 47735079979633, -191026723545976, 764362047956073, -3058170811731677
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2023

Keywords

Crossrefs

Cf. A000587, A045379, A109747, A153732, A193684, A346739, A367818.

Programs

  • Mathematica
    nmax = 25; CoefficientList[Series[Exp[1 - 4 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -4 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 25}]
Table[Sum[Binomial[n, k] (-4)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 25}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 4*x - exp(x)))) \\ Michel Marcus, Dec 02 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 4*x*A(x) - x * A(x/(1 - x)) / (1 - x).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k-4)^n / k!.
a(0) = 1; a(n) = -4*a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-4)^(n-k) * A000587(k).
Showing 1-7 of 7 results.

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