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A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).

%I A334824 #20 Sep 08 2022 08:46:25
%S A334824 1,3,0,15,0,-1,105,0,-10,0,945,0,-105,0,1,10395,0,-1260,0,21,0,135135,
%T A334824 0,-17325,0,378,0,-1,2027025,0,-270270,0,6930,0,-36,0,34459425,0,
%U A334824 -4729725,0,135135,0,-990,0,1,654729075,0,-91891800,0,2837835,0,-25740,0,55,0,13749310575,0,-1964187225,0,64324260,0,-675675,0,2145,0,-1
%N A334824 Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).
%C A334824 Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.
%H A334824 G. C. Greubel, <a href="/A334824/b334824.txt">Rows n = 0..100 of the triangle, flattened</a>
%H A334824 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 A334824 Equals the coefficients of the polynomials, g(n, x), defined by: (Start)
%F A334824 g(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k+1)!/((2*k+1)!*(n-2*k)!))*(x/2)^(n-2*k).
%F A334824 g(n, x) = ((2*n+1)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 3/2, -n, -n-1/2; -1/x^2).
%F A334824 g(n, x) = ((-i)^n/2)*(y(n+1, i*x) + (-1)^n*y(n+1, -i*x)), where y(n, x) are the Bessel Polynomials.
%F A334824 g(n, x) = (2*n-1)*x*g(n-1, x) - g(n-2, x).
%F A334824 E.g.f. of g(n, x): sin((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).
%F A334824 g(n, 1) = (-1)^n*g(n, -1) = A053984(n) = (-1)^n*A053983(-n-1) = (-1)^n*f(-n-1, 1).
%F A334824 g(n, 2) = (-1)^n*g(n, -2) = A053987(n+1). (End)
%F A334824 As a number triangle:
%F A334824 T(n, k) = i^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), where i = sqrt(-1).
%F A334824 T(n, 0) = A001147(n+1).
%e A334824 Polynomials:
%e A334824 g(0, x) = 1;
%e A334824 g(1, x) = 3*x;
%e A334824 g(2, x) = 15*x^2 - 1;
%e A334824 g(3, x) = 105*x^3 - 10*x;
%e A334824 g(4, x) = 945*x^4 - 105*x^2 + 1;
%e A334824 g(5, x) = 10395*x^5 - 1260*x^3 + 21*x;
%e A334824 g(6, x) = 135135*x^6 - 17325*x^4 + 378*x^2 - 1;
%e A334824 g(7, x) = 2027025*x^7 - 270270*x^5 + 6930*x^3 - 36*x.
%e A334824 Triangle of coefficients begins as:
%e A334824         1;
%e A334824         3, 0;
%e A334824        15, 0,      -1;
%e A334824       105, 0,     -10, 0;
%e A334824       945, 0,    -105, 0,    1;
%e A334824     10395, 0,   -1260, 0,   21, 0;
%e A334824    135135, 0,  -17325, 0,  378, 0,  -1;
%e A334824   2027025, 0, -270270, 0, 6930, 0, -36, 0.
%p A334824 T:= (n, k) -> I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!);
%p A334824 seq(seq(T(n, k), k = 0..n), n = 0..10);
%t A334824 (* First program *)
%t A334824 y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x];
%t A334824 g[n_, k_]:= Coefficient[((-I)^n/2)*(y[n+1, I*x] + (-1)^n*y[n+1, -I*x]), x, k];
%t A334824 Table[g[n, k], {n,0,10}, {k,n,0,-1}]//Flatten
%t A334824 (* Second program *)
%t A334824 Table[I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten
%o A334824 (Magma)
%o A334824 C<i> := ComplexField();
%o A334824 T:= func< n, k| Round( i^k*Factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k+1)*Factorial(n-k)) ) >;
%o A334824 [T(n,k): k in [0..n], n in [0..10]];
%o A334824 (Sage) [[ i^k*factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*factorial(k+1)*factorial(n-k)) for k in (0..n)] for n in (0..10)]
%Y A334824 Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094675, A334823.
%Y A334824 Columns k: A001147 (k=0), A000457 (k=2), A001881 (k=4), A130563 (k=6).
%K A334824 tabl,sign
%O A334824 0,2
%A A334824 _G. C. Greubel_, May 13 2020, following a suggestion from _Michel Marcus_