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 A305940 Irregular triangle where T(n,k) is the coefficient of s(y) in p(n), where s is Schur functions, p is power-sum symmetric functions, and y is the integer partition with Heinz number A215366(n,k). 4
 1, 1, -1, 1, -1, 1, 1, 0, -1, 1, -1, 1, -1, 0, 0, 1, -1, 1, 1, 0, -1, 0, 0, 1, 0, 0, -1, 1, -1, 1, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, -1, 1, 1, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 1, 0, 0, -1, 1, -1, 1, -1, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS Row n contains n nonzero entries, half of which (rounded up) are 1 and the remainder are -1. LINKS FORMULA T(n,k) = (-1)^(n - A061395(k)) if A215366(n,k) belongs to A093641 (Heinz numbers of hooks), 0 otherwise. EXAMPLE Triangle begins:   1   1 -1   1 -1  1   1  0 -1  1 -1   1 -1  0  0  1 -1  1   1  0 -1  0  0  1  0  0 -1  1 -1   1 -1  0  0  0  1  0  0  0 -1  0  0  1 -1  1   1 -1  0  0  1  0  0  0  0  0  0  0 -1  0  0  0  1  0  0 -1  1 -1 The rows correspond to the following symmetric function identities.   p(1) = s(1)   p(2) = s(2) - s(11)   p(3) = s(3) - s(21) + s(111)   p(4) = s(4) - s(31) + s(211) - s(1111)   p(5) = s(5) - s(41) + s(311) - s(2111) + s(11111) MATHEMATICA hookQ[n_]:=MatchQ[DeleteCases[FactorInteger[n], {2, _}], {}|{{_, 1}}]; Table[If[hookQ[k], (-1)^(n-Max[PrimePi/@FactorInteger[k][[All, 1]]]), 0], {n, 8}, {k, Sort[Times@@Prime/@#&/@IntegerPartitions[n]]}] CROSSREFS Cf. A056239, A082733, A093641, A153452, A153734, A215366, A296188, A296561. Cf. A304438, A305940, A317552, A317553, A317554. Sequence in context: A118172 A071039 A118111 * A119981 A115789 A212793 Adjacent sequences:  A305937 A305938 A305939 * A305941 A305942 A305943 KEYWORD sign,tabf AUTHOR Gus Wiseman, Sep 14 2018 STATUS approved

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Last modified June 24 21:50 EDT 2021. Contains 345433 sequences. (Running on oeis4.)