cp's OEIS Frontend

T(n,k) = sum(A220073(n,i): i=1..k).

Triangle read by rows in which row n lists the number of pairs of states of the subshells of the n-th shell of the nuclear shell model ordered by energy level in increasing order.

Row n lists a permutation of the first n positive integers.

If n is odd then row n lists the first (n+1)/2 odd numbers in decreasing order together with the first (n-1)/2 positive even numbers.

If n is even then row n lists the first n/2 even numbers in decreasing order together with the first n/2 odd numbers.

Row n >= 2, with its floor(n/2) last numbers taken as negative, lists the n different eigenvalues (in decreasing order) of the odd graph O(n). The odd graph O(n) has the (n-1)-subsets of a (2*n-1)-set as vertices, with two (n-1)-subsets adjacent if and only if they are disjoint. For example, O(3) is isomorphic to the Petersen graph. - Miquel A. Fiol, Apr 07 2024

From Boris Putievskiy, Jan 15 2013: (Start)

General case see A187760. Let m be natural number. Table T(n,k) n, k > 0, T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+m-1, if n < k. Table T(n,k) read by antidiagonals. The first column of the table T(n,1) is the sequence of the natural numbers A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A220073, for m=2 the result is A143182. (End)

In general, let m be natural number. T(n,k)=n-k+1, if n>=k, T(n,k)=k-n+m-1, if n A000027. In all columns with number k (k > 1) the segment with the length of (k-1): {m+k-2, m+k-3, ..., m} shifts the sequence A000027. For m=1 the result is A220073, for m=2 the result is A143182. This sequence is the result for m=3.

Row m contains m distinct elements of the set {1..m}. The last element k(m) in row m becomes the first one in row m+1. There, the remaining elements of row m repeat in the same order, with element m+1 inserted immediately after that which is the k(m)-th in row m.