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%I #21 Apr 16 2020 18:45:13
%S 1,5,25,85,275,751,1955,4615,10460,22220,45628,89420,170340,313140,
%T 562020,980628,1676370,2800410,4596290,7399930,11732006,18297950,
%U 28155910,42716750,64037980,94823756,138922300,201325900,288988100
%N a(n) has generating function 1/((1-x)^k*(1-x^2)^(k*(k-1)/2)) for k=5.
%C 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).
%H Alois P. Heinz, <a href="/A181477/b181477.txt">Table of n, a(n) for n = 0..1000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Schur_polynomial">Schur Polynomial</a>
%e 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.
%t 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]]]] ]] ]
%Y For k=2 (two variables): A002620, k=3: A038163, k=4: A054498 k=6: A181478, k=7: A181479, k=8: A181480.
%Y Column k=5 of A210391. - _Alois P. Heinz_, Mar 22 2012
%K nonn,easy
%O 0,2
%A _Wouter Meeussen_, Oct 24 2010
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