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 1-2 of 2 results.

A330797 Evaluation of the Stirling cycle polynomials at -1/2 and normalized with (-2)^n.

Original entry on oeis.org

1, 1, -1, 3, -15, 105, -945, 10395, -135135, 2027025, -34459425, 654729075, -13749310575, 316234143225, -7905853580625, 213458046676875, -6190283353629375, 191898783962510625, -6332659870762850625, 221643095476699771875, -8200794532637891559375, 319830986772877770815625
Offset: 0

Views

Author

Peter Luschny, Jan 06 2020

Keywords

Crossrefs

The equivalent for Stirling2 is A009235.

Programs

  • Magma
    m:=30;
    R:=PowerSeriesRing(Rationals(), m+2);
    A330797:= func< n | Coefficient(R!(Laplace( Sqrt(1+2*x) )), n) >;
    [A330797(n): n in [0..m]]; // G. C. Greubel, Sep 14 2023
  • Maple
    a := n -> ((-2)^(n-1)*GAMMA(n-1/2))/sqrt(Pi): seq(a(n), n=1..9);
    # Alternative:
    arec := proc(n) option remember: if n = 0 then 1 else
    (3 - 2*n)*arec(n-1) fi end: seq(arec(n), n=0..20);
    # Or:
    gf := (1+2*x)^(1/2); ser := series(gf, x, 24);
    seq(n!*coeff(ser, x, n), n=0..20);
  • Mathematica
    a[n_]:= (-2)^n*Sum[Abs[StirlingS1[n, k]]*(-1/2)^k, {k, 0, n}];
    Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 19 2021 *)
    Table[(-2)^(n-1)*Pochhammer[1/2, n-1], {n,0,30}] (* G. C. Greubel, Sep 14 2023 *)
  • SageMath
    def A330797(n): return (-2)^(n-1)*rising_factorial(1/2, n-1)
    [A330797(n) for n in (0..20)]
    

Formula

a(n) = (-2)^n*Sum_{k=0..n} |Stirling1(n,k)|*(-1/2)^k.
a(n) = (-2)^(n-1)*RisingFactorial(1/2, n-1).
a(n) = ((-2)^(n-1)*Gamma(n - 1/2))/sqrt(Pi).
a(n) = n!*[x^n] (1+2*x)^(1/2).
D-finite with recurrence a(n) = (3 - 2*n)*a(n-1).
a(n) = (-1)^(n-1)*(2*n-3)!! = (-1)^(n-1)*A001147(n-1).
a(2*n) = -2^(2*n-1)*RisingFactorial(1/2, 2*n-1) = -A103639(n-1).
a(2*n+1) = 4^n*RisingFactorial(1/2, 2*n) = A101485(n).
a(n) ~ -((-2*n)^n/exp(n))/(sqrt(2)*n).
Sum_{n>=0} 1/a(n) = 2 - sqrt(Pi/(2*e))*erfi(1/sqrt(2)), where erfi is the imaginary error function. - Amiram Eldar, Jan 08 2023
O.g.f.: 1+x*2F0(1/2,1;;-2*x). - R. J. Mathar, Aug 10 2025

A330802 Evaluation of the Big-Schröder polynomials at 1/2 and normalized with 2^n.

Original entry on oeis.org

1, 5, 33, 253, 2121, 18853, 174609, 1667021, 16290969, 162171445, 1638732129, 16765758429, 173325794409, 1807840791237, 19001320087473, 201050792435949, 2139811906460985, 22892988893079637, 246061004607915777, 2655768423781296893, 28771902274699214601
Offset: 0

Views

Author

Peter Luschny, Jan 02 2020

Keywords

Crossrefs

Programs

  • Maple
    a := proc(n) option remember; if n < 3 then return [1, 5, 33][n+1] fi;
    ((24 - 12*n)*a(n-3) + (32*n - 10)*a(n-2) + (9*n - 9)*a(n-1))/(n+1) end:
    seq(a(n), n=0..20);
    # Alternative:
    gf := 2/(1 - 4*x + sqrt(1 + 4*(x - 3)*x)):
    ser := series(gf, x, 24):
    seq(coeff(ser, x, n), n=0..20);
    # Or:
    series((x - x^2)/(3*x^2 + 4*x + 1), x, 24):
    gfun:-seriestoseries(%, 'revogf'):
    convert(%, polynom) / x: seq(coeff(%, x, n), n=0..20);
  • Mathematica
    A080247[n_, k_] := (k+1)*Sum[2^m*Binomial[n+1, m]*Binomial[n-k-1, n-k-m], {m, 0, n-k}]/(n+1);
    a[n_] := 2^n*Sum[A080247[n, k]/2^k , {k, 0, n}];
    Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 22 2023 *)
  • PARI
    N=20; x='x+O('x^N); Vec(2/(1-4*x+sqrt(1+4*(x-3)*x))) \\ Seiichi Manyama, Feb 03 2020
  • SageMath
    R. = PowerSeriesRing(QQ)
    f = (x - x^2)/(3*x^2 + 4*x + 1)
    f.reverse().shift(-1).list()
    

Formula

a(n) = 2^n*Sum_{k=0..n} A080247(n,k)/2^k.
a(n) = ((24 - 12*n)*a(n-3) + (32*n - 10)*a(n-2) + (9*n - 9)*a(n-1))/(n + 1).
a(n) = [x^n] 2/(1 - 4*x + sqrt(1 + 4*(x - 3)*x)).
a(n) = [x^n] reverse((x - x^2)/(3*x^2 + 4*x + 1))/x.
a(n) ~ 2^(n + 5/4) * (1 + sqrt(2))^(2*n-1) / (sqrt(Pi) * (57 - 40*sqrt(2)) * n^(3/2)). - Vaclav Kotesovec, Oct 22 2023
Showing 1-2 of 2 results.