ukmuiik

Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite 2 During the course of another math.se question (What's the probability of completing the illustrated "binomial walk" without ever visiting a node above the baseline?), the following generalization of $sum_i=1^n i = fracn(n+1)2$ tangentially (not directly related to that discussion) came up. Moreover, it also satisfies several "magic formulas" shown below (but not posted earlier) that suggested to me maybe it's already some well-known "thing" (i.e., "somebody's polynomial" or something). If so, anybody know what it's called, and can point me to a more complete development? Here's the definition, and then those formulas that follow from it... Define $$ left. beginarraycclcl H^1_n & = & & & 1 mbox for all $n$ \ H^2_n & = & sum_i=1^n H^1_i & = & n \ H^3_n & = & sum_i=1^n H^2_i & = & fracn(n+1)2! mbox (the u