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A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).

%I A294201 #23 Sep 20 2019 13:10:09
%S A294201 1,0,1,1,1,3,2,0,1,1,3,10,12,3,9,3,0,1,1,7,33,59,30,67,42,6,18,4,0,1,
%T A294201 1,15,106,270,216,465,420,120,235,100,10,30,5,0,1,1,31,333,1187,1365,
%U A294201 3112,3675,1596,2700,1655,330,605,195,15,45,6,0,1
%N A294201 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..3n} that are invariant under a permutation consisting of n 3-cycles (1 <= k <= 3n).
%C A294201 T(n,k) = coefficient of x^k for A(3,n)(x) in Gilbert and Riordan's article. - _Robert A. Russell_, Jun 13 2018
%H A294201 Andrew Howroyd, <a href="/A294201/b294201.txt">Table of n, a(n) for n = 1..1395</a>
%H A294201 E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.
%F A294201 T(n,k) = [n==0 & k==0] + [n>0 & k>0] * (k*T(n-1,k) + T(n-1,k-1) + T(n-1,k-3)). - _Robert A. Russell_, Jun 13 2018
%F A294201 T(n,k) = n!*[x^n*y^k] exp(Sum_{d|3} y^d*(exp(d*x) - 1)/d). - _Andrew Howroyd_, Sep 20 2019
%e A294201 Triangle begins:
%e A294201   1,  0,   1;
%e A294201   1,  1,   3,   2,   0,   1;
%e A294201   1,  3,  10,  12,   3,   9,   3,   0,   1;
%e A294201   1,  7,  33,  59,  30,  67,  42,   6,  18,   4,  0,  1;
%e A294201   1, 15, 106, 270, 216, 465, 420, 120, 235, 100, 10, 30, 5, 0, 1;
%e A294201   ...
%e A294201 Case n=2: Without loss of generality the permutation of two 3-cycles can be taken as (123)(456). The second row is [1, 1, 3, 2, 0, 1] because the set partitions that are invariant under this permutation in increasing order of number of parts are {{1, 2, 3, 4, 5, 6}}; {{1, 2, 3}, {4, 5, 6}}; {{1, 4}, {2, 5}, {3, 6}}, {{1, 5}, {2, 6}, {3, 4}}, {{1, 6}, {2, 4}, {3, 5}}; {{1, 2, 3}, {4}, {5}, {6}}, {{1}, {2}, {3}, {4, 5, 6}}, {{1}, {2}, {3}, {4}, {5}, {6}}.
%p A294201 T:= proc(n, k) option remember; `if`([n, k]=[0, 0], 1, 0)+
%p A294201      `if`(n>0 and k>0, k*T(n-1, k)+T(n-1, k-1)+T(n-1, k-3), 0)
%p A294201     end:
%p A294201 seq(seq(T(n, k), k=1..3*n), n=1..8);  # _Alois P. Heinz_, Sep 20 2019
%t A294201 T[n_, k_] := T[n,k] = If[n>0 && k>0, k T[n-1,k] + T[n-1,k-1] + T[n-1,k-3], Boole[n==0 && k==0]] (* modification of Gilbert & Riordan recursion *)
%t A294201 Table[T[n, k], {n,1,10}, {k,1,3n}] // Flatten (* _Robert A. Russell_, Jun 13 2018 *)
%o A294201 (PARI) \\ see A056391 for Polya enumeration functions
%o A294201 T(n,k)={my(ci=PermCycleIndex(CylinderPerms(3,n)[2])); StructsByCycleIndex(ci,k) - if(k>1,StructsByCycleIndex(ci,k-1))}
%o A294201 for (n=1, 6, for(k=1, 3*n, print1(T(n,k), ", ")); print);
%o A294201 (PARI)
%o A294201 G(n)={Vec(-1+serlaplace(exp(sumdiv(3, d, y^d*(exp(d*x + O(x*x^n))-1)/d))))}
%o A294201 { my(A=G(6)); for(n=1, #A, print(Vecrev(A[n]/y))) } \\ _Andrew Howroyd_, Sep 20 2019
%Y A294201 Row sums are A002874.
%Y A294201 Column k=3 gives A053156.
%Y A294201 Maximum row values are A294202.
%Y A294201 Unrelated to A002875.
%Y A294201 Cf. A002872, A002873, A293181.
%K A294201 nonn,tabf
%O A294201 1,6
%A A294201 _Andrew Howroyd_, Oct 24 2017