cp's OEIS Frontend

Maximum number of electrons in the n-th shell of an atom. - Daniel Forgues, May 09 2011

Also, column k lists the partial sums of the column k of A195151. The first differences in row n are always the n-th term of the triangular numbers repeated 0,0,1,1,3,3,6,6,... ([0,0] together with A008805).

Also, for k >= 1, this is a table of generalized polygonal numbers since column k lists the generalized m-gonal numbers, where m = k+4, for example: if k = 1 then m = 5, so the column 1 lists the generalized pentagonal numbers A001318 (see example).

Partial sums give A015128. - Omar E. Pol, Jan 09 2014

In the infinite square array if k is positive then column k is related to the generalized m-gonal numbers, where m = k+4. For example: column 1 is related to the generalized pentagonal numbers A001318. Column 2 is related to the generalized hexagonal numbers A000217 (note that A000217 is also the entry for the triangular numbers). And so on...

In the following table Euler's Pentagonal Number Theorem is represented by the entries A001318, A195310, A175003 and A000041. It seems unusual that the partition numbers are located in a middle column (see below row 1 of the table):

========================================================

. Column k of

. this square

. Generalized Triangle Triangle array A211970

k m m-gonal "A" "B" [row sums of

. numbers triangle "B"

. (if k>=1) with a(0)=1,

. if k >= 0]

========================================================

0 4 A008794 - - A211971

1 5 A001318 A195310 A175003 A000041

2 6 A000217 A195826 A195836 A006950

3 7 A085787 A195827 A195837 A036820

4 8 A001082 A195828 A195838 A195848

5 9 A118277 A195829 A195839 A195849

6 10 A074377 A195830 A195840 A195850

7 11 A195160 A195831 A195841 A195851

8 12 A195162 A195832 A195842 A195852

9 13 A195313 A195833 A195843 A196933

10 14 A195818 A210944 A210954 A210964

...

It appears that column 2 of the square array is A006950.

It appears that column 3 of the square array is A036820.

The partial sums of column 0 give A015128. - Omar E. Pol, Feb 09 2014

Rows 2n and 2n-1 both contain 1 + n^2 entries. Cf. A008794.

Row n sums to A063443(n+1).

Number of walks of length n-1 on a graph in which each node represents a 11-avoiding n-bit binary sequence B and adjacency of B and B' is determined by B'&(B|(B<<1)|(B>>1))=0 and the total number of nonzero bits in the walk is k.

Row n gives the coefficients of the independence polynomial of the n X n king graph. - Eric W. Weisstein, Jun 20 2017

A000204(n+1) = sum of n-th row, Lucas numbers;

A000204(n+3) = alternating row sum of n-th row;

A182584(n) = T(2*n,n), central terms;

A000012(n) = T(n,0), left edge;

A040000(n) = T(n,n), right edge;

A054977(n-1) = T(n,1) for n > 0;

A109613(n-1) = T(n,n-1) for n > 0;

A008794(n) = T(n,n-2) for n > 1.

For a permutation C = {c_1..c_n} of {1..n}, set s(C) = c_1 - c_2 + ... + (-1)^(n-1)*c_n. Then max s(C) is square that is (ceil(n/2))^2 or A008794(n+1).

a(n)/n! is slowly and non-monotonically decreasing: 1, 1/2, 2/3, 1/2, 3/10, 1/5, 2/7, 8/35, 23/126, 5/36, 85/462, 71/462, ... .

Positions for which a(n) divisible by all primes <= n: 1, 4, 10, ... .

The smallest primes <= n not dividing a(n) or 0 if there is no such primes: 0, 2, 3, 0, 5, 5, 7, 5, 7, 0, 7, 7, ... .

Let k = floor(n / 2). Then a(n) = divisible by k! * (n-k)!. - David A. Corneth, Oct 18 2017. (For a proof, cf. comment in A293984. - Vladimir Shevelev, Nov 06 2017)

Or row sums of the compressed triangle in A293783.

Conjecture: all terms are positive integers.

From David A. Corneth (with participation of Vladimir Shevelev), Oct 24 2017: (Start)

Conjecture is true. Proof.

1) Let C={c_1..c_n} be a permutation of {1..n}, d(C) be alternating sum c_1 - c_2 + ... +(-1)^(n-1)*c_n. Then max_{C in S_n}d(C) = A008794(n+1). Indeed, if n = 2*m, then evidently the maximum is reached on a C={2*m,1,2*m-1,2,...,m+1,m}; if n=2*m - 1, then the maximum is reached on a C={2*m-1,1,2*m-2,2,...,m-1,m}. In both cases max_{C in S_n}d(C) = m^2 = A008794(n+1). The number of distinct reaches of the maximum is, evidently, floor(n/2)!*floor((n+1)/2)! which is also Avi Peretz's representation (2001) of A010551(n). So, A293857(n) >= A010551(n) and a(n)>=1.

2) Consider two cases: a) there are no C in S_n for which d(C) = k^2 < A008794(n+1). Then A293857(n) = A010551(n) and a(n) = 1; b) there is C for which d(C) = k^2 < A008794(n+1). Then, as in 1) to reach k^2 in case n=2*m consider all (n/2)! permutations of {c_1,c_3,...,c_n} and all (n/2)! permutations of {c_2, c_4, ... , c_(n+1)}, or in case n = 2*m-1, all ((n+1)/2)! permutations of {c_1,c_3,...,c_(2*m-1)} and ((n-1)/2)! permutations of {c_2,c_4,...,c_(2*m-2)}. So we again have A010551(n) distinct reaches. If the same k^2 could be reached by another permutation C_1 (other than above permutations of C), then we again obtain A010551 distinct reaches, etc. So, A293857(n) is always divisible by A010551(n). (End)

Note that the formula mentioned in the definition gives several kinds of numbers, for example:

Row 0 and row 1 give A317300 and A317301 respectively.

Row 2 gives A001057 (canonical enumeration of integers).

Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).

Row 4 gives A008794 (squares repeated) except the initial zero.

Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).