cp's OEIS Frontend

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

%I A102728 #9 Mar 17 2018 04:15:10
%S A102728 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,
%T A102728 49,120,121,32,1,0,1,12,76,272,496,364,64,0,0,1,14,109,520,1441,2016,
%U A102728 1093,128,1,0,1,16,148,888,3376,7448,8128,3280,256,0,0,1,18,193,1400,6841
%N A102728 Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2.
%C A102728 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:
%C A102728 N: row sequence g.f. cross references.
%C A102728 0: (1^n-(-1)^n)/2 x/((1+1x)(1-1x)) A000035
%C A102728 1: (2^n-0^n)/2 x/(1-2x) A000079
%C A102728 2: (3^n-1^n)/2 x/((1-1x)(1-3x)) A003462
%C A102728 3: (4^n-2^n)/2 x/((1-2x)(1-4x)) A006516
%C A102728 4: (7^n-3^n)/2 x/((1-3x)(1-5x)) A005059
%C A102728 5: (6^n-4^n)/2 x/((1-4x)(1-6x)) A016149
%C A102728 6: (7^n-5^n)/2 x/((1-5x)(1-7x)) A016161 A081200
%C A102728 7: (8^n-6^n)/2 x/((1-6x)(1-8x)) A016170 A081201
%C A102728 8: (9^n-7^n)/2 x/((1-7x)(1-9x)) A016178 A081202
%C A102728 9: (10^n-8^n)/2 x/((1-8x)(1-10x)) A016186 A081203
%C A102728 10: (11^n-9^n)/2 x/((1-9x)(1-11x)) A016190
%C A102728 11: (12^n-10^n)/2 x/((1-10x)(1-12x)) A016196
%C A102728 ...
%C A102728 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.
%C A102728 Columns follow polynomials with certain binomial coefficients:
%C A102728 column: polynomial
%C A102728 0: 0
%C A102728 1: 1
%C A102728 2: 2n
%C A102728 3: 3n^2+ 1 (see A056107)
%C A102728 4: 4n^3+ 4n (= 8*A006003(n))
%C A102728 5: 5n^4+ 10n^2+ 1
%C A102728 6: 6n^5+ 20n^3+ 6n
%C A102728 7: 7n^6+ 35n^4+ 21n^2+ 1
%C A102728 8; 8n^7+ 56n^5+ 56n^3+ 8n
%C A102728 9: 9n^8+ 84n^6+126n^4+ 36n^2+ 1
%C A102728 10: 10n^9+ 120n^7+252n^5+120n^3+ 10n
%C A102728 11: 11n^10+165n^8+462n^6+330n^4+ 55n^2+ 1
%e A102728 Array begins:
%e A102728 0,1,0,1,0,1...
%e A102728 0,1,2,4,8,16...
%e A102728 0,1,4,13,40,121...
%e A102728 0,1,6,28,120,496...
%e A102728 0,1,8,49,272,1441...
%e A102728 ...
%o A102728 (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())
%K A102728 nonn,tabl
%O A102728 0,9
%A A102728 Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Feb 07 2005