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

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

Offset: 0

Author

Peter Luschny, Jan 02 2020

Keywords

• nonn

Links

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

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