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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.
4
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, 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, 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
OFFSET
1,2
COMMENTS
Column k gives the partial sums of the k-th column of triangle A197126.
LINKS
FORMULA
T(n,k) = Sum_{j=k..n} A197126(j,k).
T(2n+1,n+1) = A000041(n). - Alois P. Heinz, Apr 27 2018
Sum_{k=1..n} k * T(n,k) = A284870(n). - Alois P. Heinz, May 14 2018
EXAMPLE
Triangle begins:
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, 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, 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;
...
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:= proc(n, k) option remember;
b(n$2, k)[2]+`if`(n<k, 0, T(n-1, k))
end:
seq(seq(T(n, k), k=1..n), n=1..20); # Alois P. Heinz, Apr 27 2018
MATHEMATICA
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]] + If[n < k, 0, T[n-1, k]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)
CROSSREFS
Column 1 gives A000097.
Row sums give A014153.
Sequence in context: A066421 A369526 A206563 * A323954 A143983 A282988
KEYWORD
nonn,tabl
AUTHOR
Omar E. Pol, Apr 04 2018
STATUS
approved