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A304485 Regular triangle where T(n,k) is the number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions and k leaves.

%I A304485 #14 Jan 01 2021 18:12:24
%S A304485 1,1,0,1,2,0,1,6,4,0,1,12,23,7,0,1,20,81,73,12,0,1,30,209,407,206,19,
%T A304485 0,1,42,451,1566,1751,534,30,0,1,56,858,4711,9593,6695,1299,45,0,1,72,
%U A304485 1494,11951,39255,51111,23530,3004,67,0,1,90,2430,26752,130220,278570,245319,77205,6664,97,0
%N A304485 Regular triangle where T(n,k) is the number of inequivalent colorings of free pure symmetric multifunctions (with empty expressions allowed) with n positions and k leaves.
%C A304485 A free pure symmetric multifunction (with empty expressions allowed) f in EOME is either (case 1) a positive integer, or (case 2) a possibly empty expression of the form h[g_1, ..., g_k] where k >= 0, h is in EOME, each of the g_i for i = 1, ..., k is in EOME, and for i < j we have g_i <= g_j under a canonical total ordering of EOME, such as the Mathematica ordering of expressions.
%C A304485 T(n,k) is also the number of inequivalent colorings of orderless Mathematica expressions with n positions and k leaves.
%H A304485 Andrew Howroyd, <a href="/A304485/b304485.txt">Table of n, a(n) for n = 1..325</a> (rows 1..25)
%e A304485 Inequivalent representatives of the T(5,3) = 23 Mathematica expressions:
%e A304485   1[][1,1]  1[1,1][]  1[1][1]  1[1[1]]  1[1,1[]]
%e A304485   1[][1,2]  1[1,2][]  1[1][2]  1[1[2]]  1[1,2[]]
%e A304485   1[][2,2]  1[2,2][]  1[2][1]  1[2[1]]  1[2,1[]]
%e A304485   1[][2,3]  1[2,3][]  1[2][2]  1[2[2]]  1[2,2[]]
%e A304485                       1[2][3]  1[2[3]]  1[2,3[]]
%e A304485 Triangle begins:
%e A304485     1
%e A304485     1    0
%e A304485     1    2    0
%e A304485     1    6    4    0
%e A304485     1   12   23    7    0
%e A304485     1   20   81   73   12    0
%e A304485     1   30  209  407  206   19    0
%e A304485     1   42  451 1566 1751  534   30    0
%o A304485 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A304485 cycleIndexSeries(n)={my(p=O(x)); for(n=1, n, p = x*sv(1) + x*p*sExp(p)); p}
%o A304485 T(n)={my(v=Vec(InequivalentColoringsSeq(sFuncSubst(cycleIndexSeries(n), i->sv(i)*y^i)))); vector(n, n, Vecrev(v[n]/y, n))}
%o A304485 { my(A=T(10)); for(n=1, #A, print(A[n])) } \\ _Andrew Howroyd_, Jan 01 2021
%Y A304485 Row sums are A300626.
%Y A304485 Cf. A000612, A007716, A052893, A053492, A277996, A280000, A317676.
%K A304485 nonn,tabl
%O A304485 1,5
%A A304485 _Gus Wiseman_, Aug 17 2018
%E A304485 Terms a(37) and beyond from _Andrew Howroyd_, Jan 01 2021