OFFSET
1,5
COMMENTS
The k-th number in row n (with rows numbered 2,3,4,...) is the number of edges from partitions of n into k parts to partitions of n into k-1 parts, for k = n..2, where partitions p and q share an edge if p has one more part than q, and exactly one part of p is a sum of two parts of q. The limiting row is A000097, which also gives the row sums.
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..1000
EXAMPLE
First 9 rows (for n = 2 to 10):
1;
1, 1;
1, 2, 2;
1, 2, 4, 2;
1, 2, 5, 6, 3;
1, 2, 5, 8, 9, 3;
1, 2, 5, 9, 14, 12, 4;
1, 2, 5, 9, 16, 20, 16, 4;
1, 2, 5, 9, 17, 25, 30, 20, 5;
1, 2, 5, 9, 17, 27, 39, 40, 25, 5;
(See also the Example at A278762, for n = 5.)
MATHEMATICA
p[n_] := p[n] = IntegerPartitions[n];
s[n_, k_] := s[n, k] = Select[p[n], Length[#] == k &];
x[n_, k_] := x[n, k] = Map[Length, Map[Union, s[n, k]]];
b[h_] := b[h] = h (h - 1)/2;
e[n_, k_] := e[n, k] = Total[Map[b, x[n, k]]];
Flatten[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]] (* A278762 sequence *)
TableForm[Table[e[n, k], {n, 2, 20}, {k, 2, n - 1}]] (* A278762 triangle *)
Flatten[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]] (* A278763 sequence *)
TableForm[Table[e[n, k], {n, 2, 20}, {k, n - 1, 2, -1}]] (* A278763 triangle *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Nov 30 2016
STATUS
approved