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 1-5 of 5 results.

A330800 Evaluation of the Motzkin polynomials at -1/2 and normalized with (-2)^n.

Original entry on oeis.org

1, -1, 5, -17, 77, -345, 1653, -8097, 40733, -208553, 1084421, -5708785, 30370861, -163019641, 881790357, -4801746753, 26302052925, -144825094473, 801155664933, -4450426297233, 24815385947469, -138842668857369, 779247587235765, -4385948395419873, 24750623835149661
Offset: 0

Views

Author

Peter Luschny, Jan 01 2020

Keywords

Links

Crossrefs

Cf. A064189, A129400, A201641, A330799.

Programs

  • Magma
    I:=[1,-1,5]; [n le 3 select I[n] else ((6-n)*Self(n-1) + 6*(4*n-9)*Self(n-2) -36*(n-3)*Self(n-3))/n: n in [1..30]]; // G. C. Greubel, Sep 13 2023
  • Maple
    a := proc(n) option remember; if n < 3 then return [1, -1, 5][n+1] fi;
(-36*(n - 2)*a(n-3) + 6*(4*n - 5)*a(n-2) - (n - 5)*a(n-1))/(n + 1) end:
seq(a(n), n=0..24);
# Alternative:
gf := 2/(sqrt(4*x - 12*x^2 + 1) + 1):
ser := series(gf, x, 30): seq(coeff(ser,x,n), n=0..24);
# Or:
series((x^2+x)/(3*x^2+1), x, 30): gfun:-seriestoseries(%, 'revogf'):
convert(%, polynom) / x: seq(coeff(%, x, n), n=0..24);
  • Mathematica
    A330800[n_]:= Coefficient[Series[2/(Sqrt[4*x-12*x^2+1] +1), {x,0,50}], x, n]; Table[A330800[n], {n, 0, 30}] (* G. C. Greubel, Sep 13 2023 *)
  • SageMath
    R. = PowerSeriesRing(QQ)
f = (x^2 + x)/(3*x^2 + 1)
f.reverse().shift(-1).list()

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*A201641(n,k).
a(n) = (-2)^n*Sum_{k=0..n} A064189(n,k)/(-2)^k.
a(n) = (-36*(n-2)*a(n-3) + 6*(4*n-5)*a(n-2) - (n-5)*a(n-1))/(n+1).
a(n) = [x^n] 2/(sqrt(4*x - 12*x^2 + 1) + 1).
a(n) = [x^n] reverse((x^2 + x)/(3*x^2 + 1))/x.

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

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

Links

Crossrefs

Cf. A080247, A330803, A330799, A330800.

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

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.

A344507 a(n) = [x^n] 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1).

Original entry on oeis.org

1, -1, 2, -2, 5, -3, 15, 3, 59, 73, 308, 632, 1951, 4829, 13674, 36306, 100827, 275493, 765150, 2120466, 5918943, 16547595, 46452387, 130703031, 368825661, 1043125407, 2957013140, 8399389528, 23904802109, 68154435941, 194639738503, 556733127851, 1594781146419
Offset: 0

Views

Author

Peter Luschny, May 23 2021

Keywords

Links

Crossrefs

Cf. A005043, A001006, A005773, A059738, A344506, A330799.

Programs

  • Maple
    gf := 2/(3*x + sqrt((1 - 3*x)*(x + 1)) + 1):
ser := series(gf, x, 27): seq(coeff(ser, x, n), n = 0..25);
# Or:
rgf := (x - 2*x^2) / (3*x^2 - 3*x + 1):
subsop(1 = NULL, gfun:-seriestolist(series(rgf, x, 32), 'revogf'));
  • Mathematica
    a[n_] := Sum[(-2)^k Binomial[n, k] Hypergeometric2F1[(k - n)/2, (k - n + 1)/2, k + 2, 4], {k, 0, n}]; Table[a[n], {n, 0, 32}]
(* Or: *)
rgf := (x - 2 x^2) / (3 x^2 - 3 x + 1);
CoefficientList[InverseSeries[Series[rgf, {x, 0, 32}]] / x, x]
  • SageMath
    R. = PowerSeriesRing(QQ, default_prec=32)
f = (x - 2*x^2) / (3*x^2 - 3*x + 1)
f.reverse().shift(-1).list()

Formula

a(n) = [x^n] reverse((x - 2*x^2) / (3*x^2 - 3*x + 1)) / x.
a(n) = Sum_{k=0..n}(-2)^k*binomial(n, k)*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4).
a(n) = (9*(n - 2)*a(n - 3) + (12*n - 15)*a(n - 2) + (n - 5)*a(n - 1))/(2*n + 2) for n >= 3.
Showing 1-5 of 5 results.

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