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 1-6 of 6 results.

A293037 E.g.f.: exp(1 + x - exp(x)).

Original entry on oeis.org

1, 0, -1, -1, 2, 9, 9, -50, -267, -413, 2180, 17731, 50533, -110176, -1966797, -9938669, -8638718, 278475061, 2540956509, 9816860358, -27172288399, -725503033401, -5592543175252, -15823587507881, 168392610536153, 2848115497132448, 20819319685262839
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2017

Keywords

Links

Crossrefs

Column k=1 of A293051.
Column k=1 of A335977.
Cf. A000587 (k=0), this sequence (k=1), A293038 (k=2), A293039 (k=3), A293040 (k=4).
Cf. A000296, A014182, A193683, A193684, A196835.

Programs

  • Maple
    f:= series(exp(1 + x - exp(x)), x= 0, 101): seq(factorial(n) * coeff(f, x, n), n = 0..30); # Muniru A Asiru, Oct 31 2017
# second Maple program:
b:= proc(n, t) option remember; `if`(n=0, 1-2*t,
      add(b(n-j, 1-t)*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n+1, 1):
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 01 2021
  • Mathematica
    m = 26; Range[0, m]! * CoefficientList[Series[Exp[1 + x - Exp[x]], {x, 0, m}], x] (* Amiram Eldar, Jul 06 2020 *)
Table[Sum[Binomial[n, k] * BellB[k, -1], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jul 06 2020 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(-exp(x)+1+x)))
  • PARI
    a(n) = if(n==0, 1, -sum(k=0, n-2, binomial(n-1, k)*a(k))); \\ Seiichi Manyama, Aug 02 2021

Formula

a(n) = exp(1) * Sum_{k>=0} (-1)^k*(k + 1)^n/k!. - Ilya Gutkovskiy, Jun 13 2019
a(n) = Sum_{k=0..n} binomial(n,k) * Bell(k, -1). - Vaclav Kotesovec, Jul 06 2020
a(0) = 1; a(n) = - Sum_{k=0..n-2} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 2021

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).

A196835 Alternating row sums of Sheffer triangle A193685 (5-restricted Stirling2 numbers).

Original entry on oeis.org

1, 4, 15, 51, 146, 273, -319, -6374, -36235, -113833, 69388, 3772035, 28631669, 112704452, -96418909, -5652669753, -50538496446, -230554460867, 281597003109, 16303457144146, 166512491229617, 872578914956059, -1111135578108284, -78512971676777833, -919653124088665479
Offset: 0

Views

Author

Wolfdieter Lang, Oct 07 2011

Keywords

Examples

			a(2) = 25 - 11 + 1 = 15.

Links

Crossrefs

Cf. A193685, A196834 (row sums), A193683, A193684, A293037, A346740.

Programs

  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(-exp(x)+5*x+1))) \\ Michel Marcus, Aug 02 2021

Formula

a(n) = Sum_{m=0..n} (-1)^m * A193685(n,m), n>=0.
E.g.f.: exp(-exp(x)+5*x+1).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k + 5)^n / k!. - Ilya Gutkovskiy, Dec 20 2019
a(0) = 1; a(n) = 5 * a(n-1) - Sum_{k=0..n-1} binomial(n-1,k) * a(k). - Seiichi Manyama, Aug 02 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

A298373 a(n) = n! * [x^n] exp(n*x - exp(x) + 1).

Original entry on oeis.org

1, 0, 0, 1, 17, 273, 4779, 93532, 2047730, 49854795, 1339872113, 39462731031, 1265248227869, 43895994373580, 1639148060192408, 65568985769784897, 2797922570156143597, 126880981472647625557, 6094210606862471240855, 309087628703330034215088, 16508178701980033054460042
Offset: 0

Views

Author

Ilya Gutkovskiy, Jan 18 2018

Keywords

Links

Crossrefs

Cf. A000587, A074051, A134980, A193683, A193684, A196835, A290219.

Programs

  • Magma
    R:=PowerSeriesRing(Rationals(), 50);
A298373:= func< n | Coefficient(R!(Laplace( Exp(-Exp(x)+n*x+1) )), n) >;
[A298373(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-> abs(b(n, -n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 04 2021
  • Mathematica
    Table[n! SeriesCoefficient[Exp[n x - Exp[x] + 1], {x,0,n}], {n,0,20}]
Join[{1}, Table[Sum[Binomial[n, k] n^(n-k) BellB[k,-1] , {k,0,n}], {n,20}]]
  • 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) = Sum_{k=0..n} binomial(n,k)*n^(n-k)*A000587(k).
a(n) ~ exp(1-exp(1)) * n^n. - Vaclav Kotesovec, Aug 04 2021

A367818 Expansion of e.g.f. exp(1 - 3*x - exp(x)).

Original entry on oeis.org

1, -4, 15, -53, 178, -575, 1809, -5598, 17141, -52113, 157724, -475997, 1433429, -4311364, 12958627, -38909601, 116831426, -350844883, 1051414421, -3160120038, 9491592177, -28218244109, 86403627444, -255153772169, 722619907385, -2772952748516, 4627276967623, -17420488524253
Offset: 0

Views

Author

Ilya Gutkovskiy, Dec 01 2023

Keywords

Crossrefs

Cf. A000587, A005494, A109747, A153732, A193683, A346738, A367819.

Programs

  • Mathematica
    nmax = 27; CoefficientList[Series[Exp[1 - 3 x - Exp[x]], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -3 a[n - 1] - Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 27}]
Table[Sum[Binomial[n, k] (-3)^(n - k) BellB[k, -1], {k, 0, n}], {n, 0, 27}]
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(1 - 3*x - exp(x)))) \\ Michel Marcus, Dec 02 2023

Formula

G.f. A(x) satisfies: A(x) = 1 - 3*x*A(x) - x * A(x/(1 - x)) / (1 - x).
a(n) = exp(1) * Sum_{k>=0} (-1)^k * (k-3)^n / k!.
a(0) = 1; a(n) = -3*a(n-1) - Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-3)^(n-k) * A000587(k).
Showing 1-6 of 6 results.

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