Eulerian Number Ascent Calculator

Build the Eulerian row from the recurrence that inserts the newest largest value while tracking exactly k adjacent ascents. The convention counts adjacent ascents and uses k from zero through n minus one. Inputs remain whole numbers inside the displayed domain, and the page never rounds a decimal into an accepted integer. This keeps the reported result tied to one explicit mathematical definition rather than an unstated convention.

The count classifies permutations by adjacent rises, not by cycles, inversions, or blocks in a set partition. Compute the preceding row and substitute its two neighboring entries into the recurrence. Reuse the displayed recurrence, congruence, or counting identity with the same inputs. The verification evidence checks this calculator's result, but it does not establish assumptions or conclusions outside the stated mathematical domain.

The size is capped at 16 so recurrence values remain exact safe integers rather than rounded counts. The bound prevents later factorial-scale rows from losing integer precision. A finite limit prevents browser stalls and avoids presenting an unsafe floating-point integer as exact. Work beyond that limit belongs in arbitrary-precision software with explicit resource controls and independent validation.