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%I #55 Oct 29 2023 18:56:20
%S 1,1,1,3,0,1,4,2,0,1,8,2,1,0,1,11,4,2,1,0,1,19,5,3,1,1,0,1,26,10,3,3,
%T 1,1,0,1,41,11,7,3,2,1,1,0,1,56,20,8,5,3,2,1,1,0,1,83,25,13,6,5,2,2,1,
%U 1,0,1,112,38,17,11,5,5,2,2,1,1,0,1,160,49,25,13,9,5,4,2,2,1,1,0,1
%N Triangle T(n,k), n>=1, 1<=k<=n, read by rows: T(n,k) is the number of cliques of size k in all partitions of n.
%C All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.
%H Alois P. Heinz, <a href="/A197126/b197126.txt">Rows n = 1..141, flattened</a>
%H Abdulaziz M. Alanazi and Augustine O. Munagi, <a href="http://math.colgate.edu/~integers/r7/r7.Abstract.html">On partition configurations of Andrews-Deutsch</a>, Integers 17 (2017), #A7.
%F G.f. of column k: (x^k/(1-x^k)-x^(k+1)/(1-x^(k+1)))/Product_{j>0}(1-x^j).
%F Column k is asymptotic to exp(Pi*sqrt(2*n/3)) / (k*(k+1)*Pi*2^(3/2)*sqrt(n)). - _Vaclav Kotesovec_, May 24 2018
%e T(4,1) = 4: [1,1,(2)], [(1),(3)], [(4)].
%e T(8,3) = 3: [1,1,(2,2,2)], [(1,1,1),2,3], [(1,1,1),5].
%e T(12,4) = 11: [(1,1,1,1),(2,2,2,2)], [1,(2,2,2,2),3], [(1,1,1,1),2,3,3], [(3,3,3,3)], [(1,1,1,1),2,2,4], [(2,2,2,2),4], [(1,1,1,1),4,4], [(1,1,1,1),3,5], [(1,1,1,1),2,6], [(1,1,1,1),8]. Here the first partition contains 2 cliques.
%e Triangle begins:
%e 1;
%e 1, 1;
%e 3, 0, 1;
%e 4, 2, 0, 1;
%e 8, 2, 1, 0, 1;
%e 11, 4, 2, 1, 0, 1;
%e 19, 5, 3, 1, 1, 0, 1;
%e 26, 10, 3, 3, 1, 1, 0, 1;
%e ...
%p b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
%p end:
%p T:= (n, k)-> b(n, n, k)[2]:
%p seq(seq(T(n, k), k=1..n), n=1..20);
%t Table[CoefficientList[ 1/q* Tr[Flatten[q^Map[Length, Split /@ IntegerPartitions[n], {2}]]], q], {n, 24}] (* _Wouter Meeussen_, Apr 21 2012 *)
%t b[n_, p_, k_] := b[n, p, k] = If[n == 0, {1, 0}, If[p < 1, {0, 0}, Sum[ Function[l, If[m == k, l + {0, l[[1]]}, l]][b[n - p*m, p - 1, k]], {m, 0, n/p}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Aug 29 2016, after _Alois P. Heinz_ *)
%Y Columns k=1-10 give: A024786(n+1), A116646, A117524, A222704, A222705, A222706, A222707, A222708, A222709, A222710.
%Y Row sums give: A000070(n-1). Diagonal gives: A000012. Limit of reversed rows: T(2*n+1,n+1) = A002865(n).
%Y Cf. A213180.
%K nonn,tabl
%O 1,4
%A _Alois P. Heinz_, Oct 10 2011
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