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 1-3 of 3 results.

A350759 a(n) = Sum_{k=0..n} (-1)^k*A345652(k)*Stirling1(n, k).

Original entry on oeis.org

1, 0, -1, 1, 1, -4, 1, 29, -167, 1000, -7989, 75857, -794639, 9058180, -111944923, 1492748581, -21369667087, 326932765840, -5323818187817, 91947960224097, -1678914212753599, 32317295442288844, -654084630476955479, 13886774070229667213
Offset: 0

Views

Author

Mélika Tebni, Jan 14 2022

Keywords

Comments

Conjectures: For all p prime, (a(p) + a(p+1) - 2) == 0 (mod p),
a(p+1) == 1 (mod ((p+1)*p)).

Examples

			a(9) = -Sum_{k=0..7} a(k)*A238363(8, k).
a(9) = -(1*(-5040) + 0*(5760) - 1*(-3360) + 1*(1344) + 1*(-420) - 4*(112) + 1*(-28) + 29*(8)) = 1000.
E.g.f.: 1 - x^2/2! + x^3/3! + x^4/4! - 4*x^5/5! + x^6/6! + 29*x^7/7! - 167*x^8/8! + 1000*x^9/9! + ...

Crossrefs

Cf. A048994, A176118, A238363, A345652.

Programs

  • Maple
    b := proc(n) option remember; `if`(n=0, 1, add((n-1)*binomial(n-2, k)*(-1)^(n-1-k)*b(k), k=0..n-2)) end:
a := n-> add((-1)^k*b(k)*Stirling1(n, k), k=0..n):
seq(a(n), n=0..23);
# Second program:
a := proc(n) option remember; `if`(n=0, 1, add((n-2-k)!*binomial(n-1, k)*(-1)^(n-1-k)*a(k), k=0..n-2)) end:
seq(a(n), n=0..23);
# Third program:
a := series(exp(-1+(1+x)*(1-log(1+x))), x=0, 24):
seq(n!*coeff(a, x, n), n=0..23);
# Fourth program:
A350759 := n-> add(binomial(n, k)*(n-k)!*coeftayl(x^(-x), x=1, n-k), k=0..n):
seq(A350759 (n), n=0..23); # Mélika Tebni, Mar 31 2022
  • Mathematica
    CoefficientList[Series[Exp[-1+(1+x)*(1-Log[1+x])], {x, 0, 23}], x] * Range[0, 23]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(exp(-1 + (1 + x)*(1 - log(1 + x))))) \\ Michel Marcus, Jan 14 2022
  • Python
    from math import comb, factorial
def a(n):
    return 1 if n == 0 else sum([factorial(n-2-k)*comb(n-1, k)*(-1)**(n-1-k)*a(k) for k in range(n-1)])
print([a(n) for n in range(24)])

Formula

a(0) = 1, a(n) = -Sum_{k=0..n-2} a(k)*A238363(n-1, k) for n > 0.
a(0) = 1, a(n) = Sum_{k=0..n-2} (n-2-k)!*binomial(n-1, k)*(-1)^(n-1-k)*a(k) for n > 0.
E.g.f.: exp(-1 + (1 + x)*(1 - log(1 + x))).
E.g.f. y(x) satisfies y' + y*log(1 + x) = 0.
a(n) = Sum_{k=0..n} binomial(n, k)*A176118(n-k). - Mélika Tebni, Mar 31 2022
a(n) ~ -(-1)^n * n! * exp(-1) / n^2 * (1 - 2*log(n)/n). - Vaclav Kotesovec, Mar 31 2022

A346119 Expansion of the e.g.f. sqrt(2*x*exp(x) - 2*exp(x) + 3).

Original entry on oeis.org

1, 0, 1, 2, 0, -16, -35, 342, 2779, -6424, -239382, -822460, 22393657, 278844084, -1553468891, -68399947042, -275025888900, 15302175612416, 243541868882077, -2463105309082902, -121649966081262521, -473088821582805820, 50905612811064360006, 945133249101683013812, -15321255878414345388335
Offset: 0

Views

Author

Mélika Tebni, Jul 05 2021

Keywords

Examples

			sqrt(2*x*exp(x)-2*exp(x)+3) = 1 + x^2/2! + 2*x^3/3! - 16*x^5/5! - 35*x^6/6! + 342*x^7/7! + 2779*x^8/8! - 6424*x^9/9! + ...
a(11) = Sum_{k=1..5} (-1)^(k-1)*A006677(k)*A008306(11,k) = -822460.
For k=1, (-1)^(1-1)*A006677(1)*A008306(11,1) == -1 (mod 11), because A006677(1) = 1 and A008306(11,1) = (11-1)!
For k>=2, (-1)^(k-1)*A006677(k)*A008306(11,k) == 0 (mod 11), because A008306(11,k) == 0 (mod 11), result a(11) == -1 (mod 11).
a(8) = Sum_{k=1..4} (-1)^(k-1)*A006677(k)*A008306(8,k) = 2779.
a(8) == 0 (mod (8-1)), because for k >= 1, A008306(8,k) == 0 (mod 7).

Crossrefs

Cf. A000166, A006677, A008306, A327006, A345652, A345697, A345969.

Programs

  • Maple
    stirtr:= proc(p) proc(n) add(p(k)*Stirling2(n, k), k=0..n) end end: f:= n-> `if`(n=0, 1, (2*n-2)!/ (n-1)!/ 2^(n-1)): A006677:= stirtr(f): # Alois P. Heinz, 2008.
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a:= n-> add(((-1)^(k-1)*A006677(k)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(sqrt(2*x*exp(x)-2*exp(x)+3), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
  • Mathematica
    CoefficientList[Series[Sqrt(2*x*E^x-2*E^x+3), {x, 0, 24}], x] * Range[0, 24]!
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace(sqrt(2*x*exp(x) - 2*exp(x) + 3))) \\ Michel Marcus, Jul 05 2021

Formula

E.g.f. y(x) satisfies y*y' = x*exp(x).
a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A006677(k)*A008306(n,k) for n > 0.
For all p prime, a(p) == -1 (mod p).
For n > 1, a(n) == 0 (mod (n-1)).
Conjecture: a(n) = 0 for only n = 1 and n = 4.

A352376 Expansion of e.g.f. exp(1 - (1 + x) * exp(x)).

Original entry on oeis.org

1, -2, 1, 6, -2, -58, -91, 732, 4365, -1468, -140682, -685886, 1791101, 43923266, 216543097, -939472974, -22047365454, -127801626362, 541608607233, 16524264652568, 124850392700061, -279906371211584, -16968403342944782, -176737444660619046
Offset: 0

Views

Author

Seiichi Manyama, Mar 15 2022

Keywords

Links

Crossrefs

Cf. A209801, A345652.

Programs

  • Mathematica
    With[{nn=30},CoefficientList[Series[Exp[1-(1+x)Exp[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Oct 11 2024 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(1-(1+x)*exp(x))))
  • PARI
    a(n) = if(n==0, 1, -sum(k=1, n, (k+1)*binomial(n-1, k-1)*a(n-k)));

Formula

a(0) = 1; a(n) = -Sum_{k=1..n} (k+1) * binomial(n-1,k-1) * a(n-k).
Showing 1-3 of 3 results.

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