A102728 Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2.

0, 0, 1, 0, 1, 0, 0, 1, 2, 1, 0, 1, 4, 4, 0, 0, 1, 6, 13, 8, 1, 0, 1, 8, 28, 40, 16, 0, 0, 1, 10, 49, 120, 121, 32, 1, 0, 1, 12, 76, 272, 496, 364, 64, 0, 0, 1, 14, 109, 520, 1441, 2016, 1093, 128, 1, 0, 1, 16, 148, 888, 3376, 7448, 8128, 3280, 256, 0, 0, 1, 18, 193, 1400, 6841

Offset: 0

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 07 2005

Keywords

• nonn
• tabl

Comments

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

Examples

			Array begins:
0,1,0,1,0,1...
0,1,2,4,8,16...
0,1,4,13,40,121...
0,1,6,28,120,496...
0,1,8,49,272,1441...
...
		

Programs

PARI

MM(n,N)=local(M);M=matrix(n,n);for(i=1,n, for(j=1,n,if(i==j,M[i,j]=N,M[i,j]=1)));M for(k=0,12, for(i=0,k,print1((MM(2,k-i)^i)[1,2],","))) T(n, k) = ((n+1)^k-(n-1)^k)/2 for(k=0,10, for(i=0,10,print1(T(k,i),","));print()) for(k=0,10, for(i=0,10,print1(((k+1)^i-(k-1)^i)/2,","));print()) for(k=0,10, for(i=0,10,print1(polcoeff(x/((1-(k-1)*x)*(1-(k+1)*x)),i),","));print())