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

%I A330803 #13 Dec 19 2024 21:18:04
%S A330803 1,-3,17,-123,1001,-8739,79969,-756939,7349657,-72798003,732681489,
%T A330803 -7471545435,77031538377,-801616570947,8408819677377,-88821190791915,
%U A330803 943928491520249,-10085451034660947,108275140773938545,-1167408859459660923,12635538801834255401
%N A330803 Evaluation of the Big-Schröder polynomials at -1/2 and normalized with (-2)^n.
%H A330803 Seiichi Manyama, <a href="/A330803/b330803.txt">Table of n, a(n) for n = 0..941</a>
%F A330803 a(n) = (-2)^n*Sum_{k=0..n} A080247(n,k)/(-2)^k.
%F A330803 a(n) = ((8 - 4*n)*a(n-3) + (30 - 24*n)*a(n-2) + (17 - 37*n)*a(n-1))/(3*n + 3).
%F A330803 a(n) = [x^n] 2/(1 + sqrt(1 + 4*x*(x + 3))).
%F A330803 a(n) = [x^n] reverse((3*x^2 + x)/(1 - x^2))/x.
%p A330803 a := proc(n) option remember; if n < 3 then return [1, -3, 17][n+1] fi;
%p A330803 ((8 - 4*n)*a(n-3) + (30 - 24*n)*a(n-2) + (17 - 37*n)*a(n-1))/(3*n + 3) end:
%p A330803 seq(a(n), n=0..20);
%p A330803 # Alternative:
%p A330803 gf := 2/(1 + sqrt(1 + 4*x*(x + 3))):
%p A330803 ser := series(gf, x, 24):
%p A330803 seq(coeff(ser, x, n), n=0..20);
%p A330803 # Or:
%p A330803 series((3*x^2 + x)/(1 - x^2), x, 24):
%p A330803 gfun:-seriestoseries(%, 'revogf'):
%p A330803 convert(%, polynom) / x: seq(coeff(%, x, n), n=0..20);
%o A330803 (SageMath)
%o A330803 R.<x> = PowerSeriesRing(QQ)
%o A330803 f = (3*x^2 + x)/(1 - x^2)
%o A330803 f.reverse().shift(-1).list()
%o A330803 (PARI) N=20; x='x+O('x^N); Vec(2/(1+sqrt(1+4*x*(x+3)))) \\ _Seiichi Manyama_, Feb 03 2020
%Y A330803 Cf. A080247, A330802, A330799, A330800.
%K A330803 sign
%O A330803 0,2
%A A330803 _Peter Luschny_, Jan 02 2020