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

%I A330802 #15 Oct 22 2023 10:12:48
%S A330802 1,5,33,253,2121,18853,174609,1667021,16290969,162171445,1638732129,
%T A330802 16765758429,173325794409,1807840791237,19001320087473,
%U A330802 201050792435949,2139811906460985,22892988893079637,246061004607915777,2655768423781296893,28771902274699214601
%N A330802 Evaluation of the Big-Schröder polynomials at 1/2 and normalized with 2^n.
%H A330802 Seiichi Manyama, <a href="/A330802/b330802.txt">Table of n, a(n) for n = 0..941</a>
%F A330802 a(n) = 2^n*Sum_{k=0..n} A080247(n,k)/2^k.
%F A330802 a(n) = ((24 - 12*n)*a(n-3) + (32*n - 10)*a(n-2) + (9*n - 9)*a(n-1))/(n + 1).
%F A330802 a(n) = [x^n] 2/(1 - 4*x + sqrt(1 + 4*(x - 3)*x)).
%F A330802 a(n) = [x^n] reverse((x - x^2)/(3*x^2 + 4*x + 1))/x.
%F A330802 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
%p A330802 a := proc(n) option remember; if n < 3 then return [1, 5, 33][n+1] fi;
%p A330802 ((24 - 12*n)*a(n-3) + (32*n - 10)*a(n-2) + (9*n - 9)*a(n-1))/(n+1) end:
%p A330802 seq(a(n), n=0..20);
%p A330802 # Alternative:
%p A330802 gf := 2/(1 - 4*x + sqrt(1 + 4*(x - 3)*x)):
%p A330802 ser := series(gf, x, 24):
%p A330802 seq(coeff(ser, x, n), n=0..20);
%p A330802 # Or:
%p A330802 series((x - x^2)/(3*x^2 + 4*x + 1), x, 24):
%p A330802 gfun:-seriestoseries(%, 'revogf'):
%p A330802 convert(%, polynom) / x: seq(coeff(%, x, n), n=0..20);
%t A330802 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);
%t A330802 a[n_] := 2^n*Sum[A080247[n, k]/2^k , {k, 0, n}];
%t A330802 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Oct 22 2023 *)
%o A330802 (SageMath)
%o A330802 R.<x> = PowerSeriesRing(QQ)
%o A330802 f = (x - x^2)/(3*x^2 + 4*x + 1)
%o A330802 f.reverse().shift(-1).list()
%o A330802 (PARI) N=20; x='x+O('x^N); Vec(2/(1-4*x+sqrt(1+4*(x-3)*x))) \\ _Seiichi Manyama_, Feb 03 2020
%Y A330802 Cf. A080247, A330803, A330799, A330800.
%K A330802 nonn
%O A330802 0,2
%A A330802 _Peter Luschny_, Jan 02 2020