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Based on the author's previous work on the Jacobi identity for twistedrelative vertex operator algebras and modules and on the generating functionidentities for affine Lie algebras, we interpret the second difference sequenceof the sequence of the number of integer partitions with distinct parts (thestrict partitions) as the sequence of the strict partitions of n with at leastthree parts, the three largest parts consecutive, and the smallest part atleast two. The name butterfly describes both the sequence's interpretation andthe underlying bijection between the set of strict partitions of positiveintegers m with the two largest parts consecutive, and a subset of the samekind of strict partitions of m+1. Using the cyclotomic polynomials 1-x and1+x+x^2, we compute generating function identities o the butterfly sequence andrelated sequences both as infinite products and as series filtered by thenumber of parts of the corresponding partitions, and we see that the number ofpartitions of positive integers n with odd parts greater than or equal to 5 isthe sum of three consecutive terms of the butterfly sequence. We also determinea subtler merging and splitting construction of the butterfly sequence as asequence of some of the partitions with odd parts larger or equal to 3, and weoffer a related detailed interpretation of the butterfly sequence as a sequenceof what we define as generalized pentagonal, pentagonal with domino, andnon-pentagonal butterfly partitions. Finally, Euler's Pentagonal Number Theoremand a slightly different specialization of the Jacobi triple product lead torecursive algorithms to compute the butterfly sequence and related sequencesusing pentagonal number sequences and the series of triangular powers.

The butterfly sequence: the second difference sequence of the numbers of  integer partitions with distinct parts, its pentagonal number structure, its  combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2