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A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.

%I A102752 #13 Jan 18 2025 01:59:23
%S A102752 0,0,1,0,1,1,0,1,3,3,0,1,5,9,5,0,1,7,21,27,11,0,1,9,39,85,81,21,0,1,
%T A102752 11,63,203,341,243,43,0,1,13,93,405,1031,1365,729,85,0,1,15,129,715,
%U A102752 2511,5187,5461,2187,171,0,1,17,171,1157,5261,15309,25999,21845,6561,341,0,1
%N A102752 Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.
%C A102752 Consider a 3 X 3 matrix M =
%C A102752   [n, 1, 1]
%C A102752   [1, n, 1]
%C A102752   [1, 1, n].
%C A102752 The n-th row of the array contains the values of the nondiagonal elements of M^k, k=0,1,.... (Corresponding diagonal entry = nondiagonal entry + (n-1)^k.)
%C A102752 Table:
%C A102752   n: row sequence G.f. cross references.
%C A102752   0: (2^n-(-1)^n)/3 1/((1+1x)(1-2x)) A001045 (Jacobsthal sequence)
%C A102752   1: (3^n-0^n)/3 1/(1-3x) A000244
%C A102752   2: (4^n-1^n)/3 1/((1-1x)(1-4x)) A002450
%C A102752   3: (5^n-2^n)/3 1/((1-2x)(1-5x)) A016127
%C A102752   4: (6^n-3^n)/3 1/((1-3x)(1-6x)) A016137
%C A102752   5: (7^n-4^n)/3 1/((1-4x)(1-7x)) A016150
%C A102752   6: (8^n-5^n)/3 1/((1-5x)(1-8x)) A016162
%C A102752   7: (9^n-6^n)/3 1/((1-6x)(1-9x)) A016172
%C A102752   8: (10^n-7^n)/3 1/((1-7x)(1-10x)) A016181
%C A102752   9: (11^n-8^n)/3 1/((1-8x)(1-11x)) A016187
%C A102752   10:(12^n-9^n)/3 1/((1-9x)(1-12x)) A016191
%C A102752 If r(n) denotes a row sequence, r(n+1)/r(n) converges to n+2.
%C A102752 Columns follow polynomials with certain binomial coefficients:
%C A102752   column: polynomial
%C A102752   0: 0
%C A102752   1: 1
%C A102752   2: 2*n + 1
%C A102752   3: 3*n^2+ 3*n + 3
%C A102752   4: 4*n^3+ 6*n^2+ 12*n + 5
%C A102752   5: 5*n^4+10*n^3+ 30*n^2+ 25*n + 11
%C A102752   6: 6*n^5+15*n^4+ 60*n^3+ 75*n^2+ 66*n + 21
%C A102752   7: 7*n^6+21*n^5+105*n^4+ 175*n^3+ 231*n^2+ 147*n + 43
%C A102752   8: 8*n^7+28*n^6+168*n^5+ 350*n^4+ 616*n^3+ 588*n^2+344*n+ 85
%C A102752   etc.
%C A102752 Coefficients are generated by the array T(n,k)=(2^(n-k-1)-(-1)^(n-k-1))/3*(binomial(k+(n-k-1),n-k-1)) read by antidiagonals.
%e A102752 Array begins:
%e A102752   0, 1, 1,  3,   5,   11, ...
%e A102752   0, 1, 3,  9,  27,   81, ...
%e A102752   0, 1, 5, 21,  85,  341, ...
%e A102752   0, 1, 7, 39, 203, 1031, ...
%e A102752   0, 1, 9, 63, 405, 2511, ...
%e A102752   ...
%o A102752 (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=0,10, for(i=0,10,print1((MM(3,k)^i)[1,2],","));print())
%K A102752 nonn,tabl
%O A102752 0,9
%A A102752 Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Feb 09 2005