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OEIS A000338: Expansion of x^3*(5-2*x)*(1-x^3)/(1-x)^4
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In this Deep Dive we unpack OEIS A000338. We explore its generating function, explain what the offset (offset 3, 1) means, and show how the infinite power-series expansion yields the sequence beginning 5, 18, 42, 75, 117. We derive the linear recurrence and connect the terms to a combinatorial story about discordant permutations, as discussed in J. Reordan's 1954 work and traced back to N. J. Sloan’s 1973 Handbook of Integer Sequences. The episode illustrates how a compact algebraic form hides a rich mathematical landscape of counting with forbidden positions.
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