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%I #25 Aug 31 2020 03:32:10
%S 1,2,2,3,4,3,5,14,9,5,7,28,33,16,7,11,69,104,74,29,11,15,134,294,263,
%T 142,47,15,22,285,801,948,599,263,77,22,30,536,2081,3058,2425,1214,
%U 453,118,30,42,1050,5212,9769,9276,5552,2322,761,181,42,56,1918,12645,29538,34172,23770,11545,4179,1223,267,56
%N Regular triangle read rows: T(n,k) = number of non-isomorphic multiset partitions of size n and length k.
%H Andrew Howroyd, <a href="/A317533/b317533.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e Non-isomorphic representatives of the T(3,2) = 4 multiset partitions:
%e {{1},{1,1}}
%e {{1},{1,2}}
%e {{1},{2,2}}
%e {{1},{2,3}}
%e Triangle begins:
%e 1
%e 2 2
%e 3 4 3
%e 5 14 9 5
%e 7 28 33 16 7
%e 11 69 104 74 29 11
%e 15 134 294 263 142 47 15
%t permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
%t c[p_List, q_List, k_] := SeriesCoefficient[1/Product[(1 - x^LCM[p[[i]], q[[j]]])^GCD[p[[i]], q[[j]]], {j, 1, Length[q]}, {i, 1, Length[p]}], {x, 0, k}];
%t M[m_, n_, k_] := Module[{s = 0}, Do[Do[s += permcount[p]*permcount[q]*c[p, q, k], {q, IntegerPartitions[n]}], {p, IntegerPartitions[m]}]; s/(m!*n!)];
%t T[n_, k_] := M[k, n, n] - M[k - 1, n, n];
%t Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Feb 08 2020, after _Andrew Howroyd_ *)
%o (PARI) \\ See A318795 for definition of M.
%o T(n,k)={M(k, n, n) - M(k-1, n, n)}
%o for(n=1, 10, for(k=1, n, print1(T(n,k),", "));print) \\ _Andrew Howroyd_, Dec 28 2019
%o (PARI) \\ Faster version.
%o permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
%o K(q, t, n)={1/prod(j=1, #q, (1-x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))}
%o G(m,n)={my(s=0); forpart(q=m, s+=permcount(q)*exp(sum(t=1, n, (K(q, t, n)-1)/t) + O(x*x^n))); s/m!}
%o A(n,m=n)={my(p=sum(k=0, m, G(k,n)*y^k)*(1-y)); matrix(n, m, n, k, polcoef(polcoef(p, n, x), k, y))}
%o { my(T=A(10)); for(n=1, #T, print(T[n,1..n])) } \\ _Andrew Howroyd_, Aug 30 2020
%Y Row sums are A007716. First and last columns are both A000041.
%Y Cf. A034691, A255397, A255903, A255906, A317532, A334550.
%K nonn,tabl
%O 1,2
%A _Gus Wiseman_, Jul 30 2018
%E Terms a(29) and beyond from _Andrew Howroyd_, Dec 28 2019
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