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.

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

Links

Crossrefs

The equivalent for Stirling2 is A009235.
Cf. A001147, A101485, A103639, A130534.
Cf. A330799, A330800, A330802, A330803.

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

A330803 Evaluation of the Big-Schröder polynomials at -1/2 and normalized with (-2)^n.

Original entry on oeis.org

1, -3, 17, -123, 1001, -8739, 79969, -756939, 7349657, -72798003, 732681489, -7471545435, 77031538377, -801616570947, 8408819677377, -88821190791915, 943928491520249, -10085451034660947, 108275140773938545, -1167408859459660923, 12635538801834255401
Offset: 0

Views

Author

Peter Luschny, Jan 02 2020

Keywords

Links

Crossrefs

Cf. A080247, A330802, A330799, A330800.

Programs

  • Maple
    a := proc(n) option remember; if n < 3 then return [1, -3, 17][n+1] fi;
((8 - 4*n)*a(n-3) + (30 - 24*n)*a(n-2) + (17 - 37*n)*a(n-1))/(3*n + 3) end:
seq(a(n), n=0..20);
# Alternative:
gf := 2/(1 + sqrt(1 + 4*x*(x + 3))):
ser := series(gf, x, 24):
seq(coeff(ser, x, n), n=0..20);
# Or:
series((3*x^2 + x)/(1 - x^2), x, 24):
gfun:-seriestoseries(%, 'revogf'):
convert(%, polynom) / x: seq(coeff(%, x, n), n=0..20);
  • PARI
    N=20; x='x+O('x^N); Vec(2/(1+sqrt(1+4*x*(x+3)))) \\ Seiichi Manyama, Feb 03 2020
  • SageMath
    R. = PowerSeriesRing(QQ)
f = (3*x^2 + x)/(1 - x^2)
f.reverse().shift(-1).list()

Formula

a(n) = (-2)^n*Sum_{k=0..n} A080247(n,k)/(-2)^k.
a(n) = ((8 - 4*n)*a(n-3) + (30 - 24*n)*a(n-2) + (17 - 37*n)*a(n-1))/(3*n + 3).
a(n) = [x^n] 2/(1 + sqrt(1 + 4*x*(x + 3))).
a(n) = [x^n] reverse((3*x^2 + x)/(1 - x^2))/x.

A331328 Evaluation of the Little-Schröder polynomials at 1/2 and normalized with 2^n.

Original entry on oeis.org

1, 3, 21, 171, 1509, 13995, 134277, 1320651, 13237221, 134682219, 1387100229, 14430764043, 151415596197, 1600364733867, 17022016484613, 182055719885643, 1956671540189541, 21121180251536619, 228880429935661509, 2488986535173458571, 27152943714786745893
Offset: 0

Views

Author

Peter Luschny, Feb 02 2020

Keywords

Crossrefs

Cf. A172094, A001003, A330802.

Programs

  • Maple
    gf := (1+6*x-3*(4*x^2-12*x+1)^(1/2))/(30*x-2): ser := series(gf, x, 32):
seq(coeff(ser, x, n), n=0..20);
  • Mathematica
    RecurrenceTable[{a[n] == (60 a[n - 3] (n - 3) + (-184 n + 282) a[n - 2] + (27*n - 18) a[n - 1])/n, a[0] == 1, a[1] == 3, a[2] == 21}, a, {n, 20}]

Formula

a(n) = 2^n*Sum_{k=0..n} A172094(n,k) / 2^k.
a(n) = [x^n] (1 + 6*x - 3*(4*x^2 - 12*x + 1)^(1/2))/(30*x - 2).
a(n) = (60*(n - 3)*a(n-3) + (282 - 184*n)*a(n-2) + (27*n - 18)*a(n-1)) / n.
Showing 1-3 of 3 results.

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