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