A396552 - OEIS

%I #47 Jun 13 2026 09:19:22

%S 4,3,4,2,4,0,4,9,1,8,5,8,1,9,5,1,5,1,5,7,8,0,0,1,5,0,4,4,4,7,8,4,5,0,

%T 1,1,0,6,4,3,9,2,7,4,9,3,2,3,0,9,2,1,4,7,6,1,3,3,5,3,5,2,2,1,4,0,3,9,

%U 9,2,6,1,6,7,0,4,7,4,0,5,3,5,6,5,3,1,4,3,8,3,8,4,4,9,1,3,7,1,8,9,7,6,6

%N Decimal expansion of lim_{n->oo} L_n(0), where L_n is the Lebesgue function associated with the interpolation nodes 1!, 2!, ..., n!.

%C A396552 is the total variation norm of the interpolation weights l_k(0) associated with the factorial nodes 1!, 2!, 3!, ... . Equivalently, it is the operator norm of the evaluation functional at x=0 on bounded data prescribed at those nodes.

%F Equals lim_{l->oo} Sum_{k=1..l} abs(Product_{j=1..l,j<>k} -j!/(k!-j!)), from the general form: L_n(x) = Sum_{k=1..n} abs(Product_{j=1..n,j<>k} (x-j!)/(k!-j!)), with x=0. Also lim_{l->oo} Sum_{k=1..l} (Product_{j<k} j!/(k!-j!))*(Product_{j>k} j!/(j!-k!)).

%F Consider the function F(x) = Product_{k>=1} (1-x/k!), then this constant equals Sum_{k>=1} (-1)^k/(k!*F'(k!)), where F'(x) is the derivative of F(x).

%e 4.34240491858195151578... .

%o (PARI) prec = 200; default(realprecision, prec); sum(k=1, prec, abs(prod(j=1, prec, if(j==k, 1, -j!/(k!-j!)))))*1.

%Y Cf. A203306, A243257, A243258, A396533.

%K nonn,cons

%O 1,1

%A _Thomas Scheuerle_, May 29 2026