Ordered Bell Number Calculator

Start with a(0)=1, choose the nonempty first block, and sum C(n,k)a(n-k) across every possible first-block size. The indexing and counted objects are stated in the method and worked example. Inputs must be whole numbers inside the displayed domain; the page never rounds a decimal into an accepted index. This ties the answer to one explicit convention instead of silently mixing sequence offsets or combinatorial interpretations.

Ordered Bell counts block sequences, unlike ordinary Bell numbers where rearranging blocks changes nothing. Small boundary cases and the displayed identity provide useful independent checks. Recompute the displayed recurrence or closed form with the same inputs and compare its previous terms or counting factors. That check supports this bounded result, but it does not transfer the interpretation to a different sequence or counting object.

n is capped at 15 because ordered Bell numbers grow faster than ordinary set-partition counts. The implementation uses integer arithmetic internally and refuses results beyond the safe display boundary. Combinatorial and recurrence values can grow rapidly even when the inputs look small. The conservative cap prevents browser stalls and avoids presenting an unsafe floating-point integer as exact; larger work needs arbitrary-precision software and independent resource controls.