A334824 - OEIS

%I #20 Sep 08 2022 08:46:25

%S 1,3,0,15,0,-1,105,0,-10,0,945,0,-105,0,1,10395,0,-1260,0,21,0,135135,

%T 0,-17325,0,378,0,-1,2027025,0,-270270,0,6930,0,-36,0,34459425,0,

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

%C Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.

%H G. C. Greubel, <a href="/A334824/b334824.txt">Rows n = 0..100 of the triangle, flattened</a>

%H 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 Equals the coefficients of the polynomials, g(n, x), defined by: (Start)

%F 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 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 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 g(n, x) = (2*n-1)*x*g(n-1, x) - g(n-2, x).

%F E.g.f. of g(n, x): sin((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).

%F g(n, 1) = (-1)^n*g(n, -1) = A053984(n) = (-1)^n*A053983(-n-1) = (-1)^n*f(-n-1, 1).

%F g(n, 2) = (-1)^n*g(n, -2) = A053987(n+1). (End)

%F As a number triangle:

%F 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 T(n, 0) = A001147(n+1).

%e Polynomials:

%e g(0, x) = 1;

%e g(1, x) = 3*x;

%e g(2, x) = 15*x^2 - 1;

%e g(3, x) = 105*x^3 - 10*x;

%e g(4, x) = 945*x^4 - 105*x^2 + 1;

%e g(5, x) = 10395*x^5 - 1260*x^3 + 21*x;

%e g(6, x) = 135135*x^6 - 17325*x^4 + 378*x^2 - 1;

%e g(7, x) = 2027025*x^7 - 270270*x^5 + 6930*x^3 - 36*x.

%e Triangle of coefficients begins as:

%e         1;

%e         3, 0;

%e        15, 0,      -1;

%e       105, 0,     -10, 0;

%e       945, 0,    -105, 0,    1;

%e     10395, 0,   -1260, 0,   21, 0;

%e    135135, 0,  -17325, 0,  378, 0,  -1;

%e   2027025, 0, -270270, 0, 6930, 0, -36, 0.

%p T:= (n, k) -> I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!);

%p seq(seq(T(n, k), k = 0..n), n = 0..10);

%t (* First program *)

%t y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x];

%t g[n_, k_]:= Coefficient[((-I)^n/2)*(y[n+1, I*x] + (-1)^n*y[n+1, -I*x]), x, k];

%t Table[g[n, k], {n,0,10}, {k,n,0,-1}]//Flatten

%t (* Second program *)

%t 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 (Magma)

%o C<i> := ComplexField();

%o 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 [T(n,k): k in [0..n], n in [0..10]];

%o (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 Cf. A001497, A001498, A053983, A053984, A053987, A053988, A094675, A334823.

%Y Columns k: A001147 (k=0), A000457 (k=2), A001881 (k=4), A130563 (k=6).

%K tabl,sign

%O 0,2

%A _G. C. Greubel_, May 13 2020, following a suggestion from _Michel Marcus_