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A102728 Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2. 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
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
LINKS
EXAMPLE
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...
...
PROG
(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())
CROSSREFS
Sequence in context: A238889 A296207 A253628 * A262495 A352687 A336703
KEYWORD
nonn,tabl
AUTHOR
Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 07 2005
STATUS
approved

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Last modified April 20 15:50 EDT 2024. Contains 371844 sequences. (Running on oeis4.)