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 A317533 Regular triangle read rows: T(n,k) = number of non-isomorphic multiset partitions of size n and length k. 60
 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, 142, 47, 15, 22, 285, 801, 948, 599, 263, 77, 22, 30, 536, 2081, 3058, 2425, 1214, 453, 118, 30, 42, 1050, 5212, 9769, 9276, 5552, 2322, 761, 181, 42, 56, 1918, 12645, 29538, 34172, 23770, 11545, 4179, 1223, 267, 56 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows) EXAMPLE Non-isomorphic representatives of the T(3,2) = 4 multiset partitions:   {{1},{1,1}}   {{1},{1,2}}   {{1},{2,2}}   {{1},{2,3}} Triangle begins:     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  142   47   15 MATHEMATICA 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]; 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}]; 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[n_, k_] := M[k, n, n] - M[k - 1, n, n]; Table[T[n, k], {n, 1, 11}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 08 2020, after Andrew Howroyd *) PROG (PARI) \\ See A318795 for definition of M. T(n, k)={M(k, n, n) - M(k-1, n, n)} for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print) \\ Andrew Howroyd, Dec 28 2019 (PARI) \\ Faster version. 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} K(q, t, n)={1/prod(j=1, #q, (1-x^lcm(t, q[j]) + O(x*x^n))^gcd(t, q[j]))} 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!} 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))} { my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Aug 30 2020 CROSSREFS Row sums are A007716. First and last columns are both A000041. Cf. A034691, A255397, A255903, A255906, A317532, A334550. Sequence in context: A316939 A259478 A155706 * A231227 A231441 A284199 Adjacent sequences:  A317530 A317531 A317532 * A317534 A317535 A317536 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Jul 30 2018 EXTENSIONS Terms a(29) and beyond from Andrew Howroyd, Dec 28 2019 STATUS approved

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Last modified April 20 01:55 EDT 2021. Contains 343118 sequences. (Running on oeis4.)