# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a317449 Showing 1-1 of 1 %I A317449 #34 Dec 30 2020 14:57:14 %S A317449 1,2,2,3,6,3,5,21,16,5,7,52,72,32,7,11,141,306,216,65,11,15,327,1113, %T A317449 1160,512,113,15,22,791,4033,6052,3737,1154,199,22,30,1780,13586, %U A317449 28749,24325,10059,2317,323,30,42,4058,45514,133642,151994,82994,24854,4493,523,42 %N A317449 Regular triangle where T(n,k) is the number of multiset partitions of strongly normal multisets of size n into k blocks, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities. %H A317449 Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50) %e A317449 The T(3,2) = 6 multiset partitions are {{1},{1,1}}, {{1},{1,2}}, {{2},{1,1}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}. %e A317449 Triangle begins: %e A317449 1 %e A317449 2 2 %e A317449 3 6 3 %e A317449 5 21 16 5 %e A317449 7 52 72 32 7 %e A317449 11 141 306 216 65 11 %e A317449 15 327 1113 1160 512 113 15 %e A317449 ... %t A317449 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}]; %t A317449 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]]; %t A317449 strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]; %t A317449 Table[Length[Select[Join@@mps/@strnorm[n],Length[#]==k&]],{n,6},{k,n}] %o A317449 (PARI) %o A317449 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)} %o A317449 D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1, -n)/prod(i=1, #v, i^v[i]*v[i]!)} %o A317449 U(m, n)={my(s=0); forpart(p=m, s+=D(p,n)); s} %o A317449 M(n)={Mat(vector(n,k,(U(k,n)-U(k-1,n))~))} %o A317449 { my(A=M(8)); for(n=1, #A~, print(A[n,1..n])) } \\ _Andrew Howroyd_, Dec 30 2020 %Y A317449 Row sums are A035310. First and last columns are both A000041. %Y A317449 Cf. A001055, A007716, A045778, A255906, A281116, A317584, A317654, A317755, A317775, A317776. %K A317449 nonn,tabl %O A317449 1,2 %A A317449 _Gus Wiseman_, Aug 06 2018 %E A317449 Terms a(46) and beyond from _Andrew Howroyd_, Dec 30 2020 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE