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A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).

%I A334823 #22 Sep 08 2022 08:46:25
%S A334823 1,1,0,3,0,-1,15,0,-6,0,105,0,-45,0,1,945,0,-420,0,15,0,10395,0,-4725,
%T A334823 0,210,0,-1,135135,0,-62370,0,3150,0,-28,0,2027025,0,-945945,0,51975,
%U A334823 0,-630,0,1,34459425,0,-16216200,0,945945,0,-13860,0,45,0,654729075,0,-310134825,0,18918900,0,-315315,0,1485,0,-1
%N A334823 Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).
%C A334823 Lambert's numerator polynomials related to convergents of tan(x), g(n, x), are given in A334824.
%H A334823 G. C. Greubel, <a href="/A334823/b334823.txt">Rows n = 0..100 of the triangle, flattened</a>
%H A334823 J.-H. Lambert, <a href="http://www.kuttaka.org/~JHL/L1768b.pdf">Mémoire sur quelques propriétés remarquables des quantités transcendantes et logarithmiques</a> (Memoir on some properties that can be traced from circular transcendent and logarithmic quantities), Histoire de l’Académie royale des sciences et belles-lettres (1761), Berlin. See <a href="http://www.bibnum.education.fr/mathematiques/theorie-des-nombres/lambert-et-l-irrationalite-de-p-1761">also</a>.
%F A334823 Equals the coefficients of the polynomials, f(n, x), defined by: (Start)
%F A334823 f(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k)!/((2*k)!*(n-2*k)!))*(x/2)^(n-2*k).
%F A334823 f(n, x) = ((2*n)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 1/2, -n, -n+1/2; -1/x^2).
%F A334823 f(n, x) = ((-i)^n/2)*(y(n, i*x) + (-1)^n*y(n, -i*x)), where y(n, x) are the Bessel Polynomials.
%F A334823 f(n, x) = (2*n-1)*x*f(n-1, x) - f(n-2, x).
%F A334823 E.g.f. of f(n, x): cos((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).
%F A334823 f(n, 1) = (-1)^n*f(n, -1) = A053983(n) = (-1)^(n+1)*A053984(-n-1) = (-1)^(n+1) * g(-n-1, 1).
%F A334823 f(n, 2) = (-1)^n*f(n, -2) = A053988(n+1). (End)
%F A334823 As a number triangle:
%F A334823 T(n, k) = i^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), where i = sqrt(-1).
%F A334823 T(n, 0) = A001147(n).
%e A334823 Polynomials:
%e A334823 f(0, x) = 1;
%e A334823 f(1, x) = x;
%e A334823 f(2, x) = 3*x^2 - 1;
%e A334823 f(3, x) = 15*x^3 - 6*x;
%e A334823 f(4, x) = 105*x^4 - 45*x^2 + 1;
%e A334823 f(5, x) = 945*x^5 - 420*x^3 + 15*x;
%e A334823 f(6, x) = 10395*x^6 - 4725*x^4 + 210*x^2 - 1;
%e A334823 f(7, x) = 135135*x^7 - 62370*x^5 + 3150*x^3 - 28*x;
%e A334823 f(8, x) = 2027025*x^8 - 945945*x^6 + 51975*x^4 - 630*x^2 + 1.
%e A334823 Triangle of coefficients begins as:
%e A334823         1;
%e A334823         1, 0;
%e A334823         3, 0,      -1;
%e A334823        15, 0,      -6, 0;
%e A334823       105, 0,     -45, 0,     1;
%e A334823       945, 0,    -420, 0,    15, 0;
%e A334823     10395, 0,   -4725, 0,   210, 0,   -1;
%e A334823    135135, 0,  -62370, 0,  3150, 0,  -28, 0;
%e A334823   2027025, 0, -945945, 0, 51975, 0, -630, 0, 1.
%p A334823 T:= (n, k) -> I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!);
%p A334823 seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10);
%t A334823 (* First program *)
%t A334823 y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x];
%t A334823 f[n_, k_]:= Coefficient[((-I)^n/2)*(y[n, I*x] + (-1)^n*y[n, -I*x]), x, k];
%t A334823 Table[f[n, k], {n,0,10}, {k,n,0,-1}]//Flatten
%t A334823 (* Second program *)
%t A334823 Table[ I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten
%o A334823 (Magma)
%o A334823 C<i> := ComplexField();
%o A334823 T:= func< n, k| Round( i^k*Factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k)*Factorial(n-k)) ) >;
%o A334823 [T(n,k): k in [0..n], n in [0..10]];
%o A334823 (Sage) [[ i^k*factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*factorial(k)*factorial(n-k)) for k in (0..n)] for n in (0..10)]
%Y A334823 Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094674, A334824.
%Y A334823 Columns k: A001147 (k=0), A001879 (k=2), A001880 (k=4), A038121 (k=6).
%K A334823 tabl,sign
%O A334823 0,4
%A A334823 _G. C. Greubel_, May 12 2020, following a suggestion from _Michel Marcus_