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A299779 Triangle read by rows: T(n,k) is the total number of cliques of size k in all partitions of all positive integers <= n.

%I A299779 #25 Dec 07 2019 12:06:08
%S A299779 1,2,1,5,1,1,9,3,1,1,17,5,2,1,1,28,9,4,2,1,1,47,14,7,3,2,1,1,73,24,10,
%T A299779 6,3,2,1,1,114,35,17,9,5,3,2,1,1,170,55,25,14,8,5,3,2,1,1,253,80,38,
%U A299779 20,13,7,5,3,2,1,1,365,118,55,31,18,12,7,5,3,2,1,1,525,167,80,44,27,17,11,7,5,3,2,1,1
%N A299779 Triangle read by rows: T(n,k) is the total number of cliques of size k in all partitions of all positive integers <= n.
%C A299779 Column k gives the partial sums of the k-th column of triangle A197126.
%H A299779 Alois P. Heinz, <a href="/A299779/b299779.txt">Rows n = 1..150, flattened</a>
%F A299779 T(n,k) = Sum_{j=k..n} A197126(j,k).
%F A299779 T(2n+1,n+1) = A000041(n). - _Alois P. Heinz_, Apr 27 2018
%F A299779 Sum_{k=1..n} k * T(n,k) = A284870(n). - _Alois P. Heinz_, May 14 2018
%e A299779 Triangle begins:
%e A299779     1;
%e A299779     2,   1;
%e A299779     5,   1,  1;
%e A299779     9,   3,  1,  1;
%e A299779    17,   5,  2,  1,  1;
%e A299779    28,   9,  4,  2,  1,  1;
%e A299779    47,  14,  7,  3,  2,  1,  1;
%e A299779    73,  24, 10,  6,  3,  2,  1,  1;
%e A299779   114,  35, 17,  9,  5,  3,  2,  1,  1;
%e A299779   170,  55, 25, 14,  8,  5,  3,  2,  1,  1;
%e A299779   253,  80, 38, 20, 13,  7,  5,  3,  2,  1,  1;
%e A299779   365, 118, 55, 31, 18, 12,  7,  5,  3,  2,  1,  1;
%e A299779   525, 167, 80, 44, 27, 17, 11,  7,  5,  3,  2,  1,  1;
%e A299779 ...
%p A299779 b:= proc(n, p, k) option remember; `if`(n=0, [1, 0], `if`(p<1, [0, 0],
%p A299779       add((l->`if`(m=k, l+[0, l[1]], l))(b(n-p*m, p-1, k)), m=0..n/p)))
%p A299779     end:
%p A299779 T:= proc(n, k) option remember;
%p A299779       b(n$2, k)[2]+`if`(n<k, 0, T(n-1, k))
%p A299779     end:
%p A299779 seq(seq(T(n, k), k=1..n), n=1..20);  # _Alois P. Heinz_, Apr 27 2018
%t A299779 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 A299779 T[n_, k_] := b[n, n, k][[2]] + If[n < k, 0, T[n-1, k]];
%t A299779 Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Dec 07 2019, after _Alois P. Heinz_ *)
%Y A299779 Column 1 gives A000097.
%Y A299779 Row sums give A014153.
%Y A299779 Cf. A000041, A000070, A197126, A284870.
%K A299779 nonn,tabl
%O A299779 1,2
%A A299779 _Omar E. Pol_, Apr 04 2018