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Consider some substring S[i...j]. Let parity[K] for 0<=K<=9 be the parity of the number of occurrences of digit K in subtring S[i...j]. Notice that the substring will be a palindrome if and only if parity[K] is 0 for all K or there is only one K for which parity[K] is 1 and parity[K] is 0 for all other Ks.

Having that in mind, we will loop from 1 to N and at each step we will find the number of those palindromes ending at the current position i. In order to do that, let parity[K] be the described parity for digit K in substring S[1...i]. Consider some j (1<=j<=i). Substring S[j...i] can be made a palindrome if all parities for position j-1 are the same as the parities for position i or there is only one difference.

For each position, we can store the parities as a bitmask with 10 bits and that way we can find the answer quickly. Check my code for more details: http://ideone.com/1grO2A :)