Unsigned Stirling Cycle Calculator

Build unsigned first-kind Stirling rows from c(n,k) = c(n-1,k-1) + (n-1)c(n-1,k), starting with c(0,0) = 1. The unsigned convention reports a count rather than the alternating sign used by one algebraic notation. 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.

The value classifies permutations by disjoint cycle count, not set partitions into unlabeled blocks. The previous row supplies the new-cycle and insertion terms. 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 18 so every permitted cycle count remains an exact safe integer. The cap also keeps every intermediate row exact. 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.