Counting pure equilibria in a sealed coin-chest auction

Ten bidders compete for a chest whose dollar value equals the number of heads in $20$ fair coin tosses made out of sight; everyone knows this rule but nobody sees the tosses. Each bidder submits a bid that is a non-negative whole number of dollars. A highest bidder (chosen uniformly at random if several tie) receives the chest and pays their bid; all other bids are refunded. Bidders are risk-neutral (utility is linear in money over the relevant range of $0 to $10). How many pure-strategy Nash equilibria (bid profiles where every player bids one value with certainty) does this game have?

$5{,}937{,}424{,}602$. The chest is worth $10 in expectation, so no one bids above 10. If two or more bid 10, everyone's payoff is $0$ and no deviation helps, whatever the others bid in $\{0,\dots,10\}$: that gives $11^{10}-10^{10}-10\cdot10^{9}$ profiles. Otherwise the only equilibrium is everyone bidding 9 (a lone 10-bidder would drop to 9, and any lower top bid invites outbidding), adding $1$.