cp's OEIS Frontend

Also, this is the number of incongruent integer-edged Heron triangles whose circumdiameter is the product of n distinct primes each of shape 4k + 1. Cf. A003462, A109021. - R. K. Guy, Jan 31 2007

Binomial transform of A081200.

Conjecture (verified up to a(8)): Number of collinear 6-tuples of points in a 6 X 6 X 6 X... n-dimensional cubic grid. [R. H. Hardin, May 23 2010]

From Wolfdieter Lang, Jul 17 2017: (Start)

For a combinatorial interpretation of a(n) with special 8-letter words of length n see the comment in A081200 on the 7-letter analog.

The binomial transform of {a(n)}_{n >= 0} is A081202, the 9-letter analog.

(End)

Binomial transform of A005059.

Conjecture (verified up to a(9)): Number of collinear 4-tuples of points in a 4 X 4 X 4 X ... n-dimensional cubic grid. - R. H. Hardin, May 24 2010

a(n) is also the total number of words of length n, over an alphabet of six letters, of which one of them appears an odd number of times. See a Lekraj Beedassy, Jul 22 2003, comment on A006516 (4-letter case), and the Balakrishnan reference there. For the 2-, 3-, 5- and 7-letter case analogs see A131577, A003462, A005059 and A081200, respectively. - Wolfdieter Lang, Jul 16 2017

Binomial transform of A081201.

From Wolfdieter Lang, Jul 17 2017: (Start)

For a combinatorial interpretation of a(n) with special 9-letter words of length n see the comment in A081200 on the 7-letter analog.

The binomial transform of {a(n)}_{n >=0} is A081203, the 10-letter analog.

(End)

Binomial transform of A081202.

From Wolfdieter Lang, Jul 17 2017: (Start)

For a combinatorial interpretation of a(n) with special 10-letter words of length n see the comment in A081200 on the 7-letter analog.

The binomial transform of {a(n)}_{n >= 0} is {0, A016190}, the 11-letter analog.

(End)

Comment from Peter Bala (Dec 06 2011): "Let P denote Pascal's triangle A070318 and put M = 1/2*(P-P^-1). M is A162590 (see also A131047). Then the first column of (I-t*M)^-1 (apart from the initial 1) lists the row polynomials for" A196776(n,k), which gives the number of ordered partitions of an n set into k odd-sized blocks. - Peter Luschny, Dec 06 2011

The n-th row of the triangle is formed by multiplying by 2^(n-1) the elements of the first row of the limit as k approaches infinity of the stochastic matrix P^(2k-1) where P is the stochastic matrix associated with the Ehrenfest model with n balls. The elements of a stochastic matrix P give the probability of arriving in a state j given the previous state i. In particular the sum of every row of the matrix must be 1, and so the sum of the terms in the n-th row of this triangle is 2^(n-1). Furthermore, by the properties of Markov chains, we can interpret P^(2k) as the (2k)-step transition matrix of the Ehrenfest model and its limit exists and it is again a stochastic matrix. The rows of the triangle divided by 2^(n-1) are the even rows (second, fourth, ...) and the odd rows (first, third, ...) of the limit matrix P^(2k). - Luca Onnis, Oct 29 2023

For every choice of integers r and s, the polynomials p(n,x) form a strong divisibility sequence. Thus, if r, s, and x are integers, then p(x,n) is a strong divisibility sequence. That is, gcd(p(x,h),p(x,k)) = p(x,gcd(h,k)).

Consider a 2 X 2 matrix M = [N, 1] / [1, N]. The n-th row of the array contains the values of the non-diagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = non-diagonal entry + (N-1)^k.) Table:

N: row sequence g.f. cross references.

0: (1^n-(-1)^n)/2 x/((1+1x)(1-1x)) A000035

1: (2^n-0^n)/2 x/(1-2x) A000079

2: (3^n-1^n)/2 x/((1-1x)(1-3x)) A003462

3: (4^n-2^n)/2 x/((1-2x)(1-4x)) A006516

4: (7^n-3^n)/2 x/((1-3x)(1-5x)) A005059

5: (6^n-4^n)/2 x/((1-4x)(1-6x)) A016149

6: (7^n-5^n)/2 x/((1-5x)(1-7x)) A016161 A081200

7: (8^n-6^n)/2 x/((1-6x)(1-8x)) A016170 A081201

8: (9^n-7^n)/2 x/((1-7x)(1-9x)) A016178 A081202

9: (10^n-8^n)/2 x/((1-8x)(1-10x)) A016186 A081203

10: (11^n-9^n)/2 x/((1-9x)(1-11x)) A016190

11: (12^n-10^n)/2 x/((1-10x)(1-12x)) A016196

...

Characteristic polynomial x^2-2nx+(n^2-1) has roots n+-1, so if r(n) denotes a row sequence, r(n+1)/r(n) converges to n+1.

Columns follow polynomials with certain binomial coefficients:

column: polynomial

0: 0

1: 1

2: 2n

3: 3n^2+ 1 (see A056107)

4: 4n^3+ 4n (= 8*A006003(n))

5: 5n^4+ 10n^2+ 1

6: 6n^5+ 20n^3+ 6n

7: 7n^6+ 35n^4+ 21n^2+ 1

8; 8n^7+ 56n^5+ 56n^3+ 8n

9: 9n^8+ 84n^6+126n^4+ 36n^2+ 1

10: 10n^9+ 120n^7+252n^5+120n^3+ 10n

11: 11n^10+165n^8+462n^6+330n^4+ 55n^2+ 1