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

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

Offset: 0

Author

Peter Luschny, Jan 02 2020

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..941

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.