The origin of these sequnces is in this. Let's examine the following recurrences (and their MCPs):

F1 = 0,1,0,1,0,1,0,1,0,1,...
MCP of F1 = x^2 - 1

F2 = 0,3,0,7,0,11,0,15,0,19,0,23,0,27,0,31
MCP of F2 = x^4 - 2*x^2 + 1

F3 = 0,6,3,31,10,76,21,141,36,226,55,331,78,456,105,601
MCP of F3 = x^6 - 3*x^4 + 3*x^2 - 1

F4 = 0,10,21,117,122,448,367,1131,820,2294,1545,4065,2606,6572,4067,9943
MCP of F4 = x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1

F5 = 0,16,89,439,906,2630,3907,9037,11380,23196,26445,49731,53054,94354,95991,163865
MCP of F5 = x^10 - 5*x^8 + 10*x^6 - 10*x^4 + 5*x^2 - 1

F6 = 0,28,307,1729,5478,16126,33865,75299,128588,244416,368543,633765,879410,1410786,1844661,2812103,3516568,5155556,6227211,8852233,10399486,14418502,16558113,22488043,25340644,33823880,37508471,49330413,53957834,70065450
MCP of F6 = x^12 - 6*x^10 + 15*x^8 - 20*x^6 + 15*x^4 - 6*x^2 + 1

etc...

It is obvious that the coefficinets of MCP of Fn equal terms of n-th row of Pascal's triangle.

Suppose I want to calculate some terms of reccurence F100. I can easily determine coefficinets of linear reccurence but I also need to calculate initial values. So I found that they (initial values) also can be defined by recurrences for which I asked in a Stackoverflow's question.