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 A197126 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. 13
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. LINKS Alois P. Heinz, Rows n = 1..141, flattened Abdulaziz M. Alanazi, Augustine O. Munagi, On partition configurations of Andrews-Deutsch, Integers 17 (2017), #A7. FORMULA G.f. of column k: (x^k/(1-x^k)-x^(k+1)/(1-x^(k+1)))/Product_{j>0}(1-x^j). 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 EXAMPLE T(4,1) = 4: [1,1,(2)], [(1),(3)], [(4)]. T(8,3) = 3: [1,1,(2,2,2)], [(1,1,1),2,3], [(1,1,1),5]. 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. Triangle begins:    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, 1, 1, 0, 1; MAPLE b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],       add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))     end: T:= (n, k)-> b(n, n, k)[2]: seq(seq(T(n, k), k=1..n), n=1..20); MATHEMATICA Table[CoefficientList[ 1/q* Tr[Flatten[q^Map[Length, Split /@ IntegerPartitions[n], {2}]]], q], {n, 24}] (* Wouter Meeussen, Apr 21 2012 *) 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 *) CROSSREFS Columns k=1-10 give: A024786(n+1), A116646, A117524, A222704, A222705, A222706, A222707, A222708, A222709, A222710. Row sums give: A000070(n-1). Diagonal gives: A000012.  Limit of reversed rows: T(2*n+1,n+1) = A002865(n). Cf. A213180. Sequence in context: A245120 A226912 A177330 * A256987 A048963 A119458 Adjacent sequences:  A197123 A197124 A197125 * A197127 A197128 A197129 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Oct 10 2011 STATUS approved

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Last modified January 24 19:19 EST 2020. Contains 331211 sequences. (Running on oeis4.)