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A181477
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a(n) has generating function 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)) for k=5.
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5
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1, 5, 25, 85, 275, 751, 1955, 4615, 10460, 22220, 45628, 89420, 170340, 313140, 562020, 980628, 1676370, 2800410, 4596290, 7399930, 11732006, 18297950, 28155910, 42716750, 64037980, 94823756, 138922300, 201325900, 288988100
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OFFSET
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0,2
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COMMENTS
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a(n-1,k) is conjectured to also be the count of monomials (or terms) in the Schur polynomials of k variables and degree n, summed over all partitions of n in at most k parts (zero-padded to length k).
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LINKS
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EXAMPLE
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a(3)=85 since the Schur polynomial of 5 variables and degree 4 starts off as x[1]*x[2]*x[3]*x[4] + x[1]*x[2]*x[3]*x[5] + ... + x[4]*x[5]^3 + x[5]^4. The exponents collect to the padded partitions of 4 as 5*p(1) + 40*p(2) + 30*p(3) + 150*p(4) + 50*p(5) where p(1) is the lexicographically first padded partition of 4: {4,0,0,0}, a coded form of monomials x[i]^4, and p(5) stands for {1,1,1,1}, coding x[i]x[j]x[k]x[l] with all indices different.
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MATHEMATICA
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Tr[toz/@(Function[q, PadRight[q, k]]/@ (TransposePartition/@ Partitions[n, k]))/. w[arg__] -> 1 ]; with toz[p_]:=Block[{a, q, e, w}, u1=Expand[q Together[Expand[schur[p]]] +q a]/. Plus-> List ; u2=u1/. Times->w /. q->Sequence[]/. w[i_Integer, r__]-> i w[r] /. x[_]^(e_:1) ->e ; u3=Plus@@ u2/. w[arg__]:> Reverse@ Sort@ w[arg] /. w[a]->0 ]; and schur[p_]:=Block[{le=Length[p], n=Tr[p]}, Together[Expand[Factor[Det[Outer[ #2^#1&, p+le-Range[le] , Array[x, le]]]]/Factor[Det[Outer[ #2^#1&, Range[le-1, 0, -1] , Array[x, le]]]] ]] ]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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