A396552 Decimal expansion of lim_{n->oo} L_n(0), where L_n is the Lebesgue function associated with the interpolation nodes 1!, 2!, ..., n!.

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, 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, 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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..103.

Formula

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!)).

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).

Example

4.34240491858195151578... .

Prog

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

Crossrefs

Cf. A203306, A243257, A243258, A396533.

Sequence in context: A266110 A204819 A161882 * A276789 A082125 A058290

Adjacent sequences: A396549 A396550 A396551 * A396553 A396554 A396555

Keyword

nonn,cons

Author

Thomas Scheuerle, May 29 2026

Status

approved