Triangle of the number of palindromic (in bases 2 to n) quadruples that sum to n.

1, 2, 2, 2, 4, 4, 2, 3, 5, 5, 3, 4, 6, 8, 8, 3, 3, 6, 8, 10, 10, 4, 5, 7, 10, 12, 14, 14, 4, 3, 6, 10, 13, 15, 17, 17, 6, 6, 7, 11, 15, 18, 20, 22, 22, 4, 4, 6, 9, 15, 19, 22, 24, 26, 26, 7, 7, 7, 12, 16, 22, 26, 29, 31, 33, 33, 4, 6, 6, 9, 15, 21, 27, 31, 34

1

It appears that all terms are positive. See S000744 and S000743 for sums of 2 and 3 terms.

T. D. Noe, Plot of 49 rows

T. D. Noe, Table of 49 rows

Yu Gao, Represent a natural number as the sum of palindromes in various bases, arXiv 1508.06185 (Aug 24 2015)

(Mma) nn = 20; pal = Table[{}, {nn}]; Do[pal[[b]] = Select[Range[0, nn], (d = IntegerDigits[#, b]; d == Reverse[d]) &], {b, 2, nn}]; Table[Table[Length[IntegerPartitions[n, {4}, pal[[b]]]], {b, 2, n}], {n, 2, nn}]

nonn,base,tabl

T. D. Noe, Nov 02 2015