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%I #18 Nov 12 2019 04:16:06
%S 1,0,0,0,1,0,0,1,1,0,0,0,2,0,0,0,0,2,2,0,0,0,0,1,5,1,0,0,0,0,0,5,5,0,
%T 0,0,0,0,0,3,11,3,0,0,0,0,0,0,1,13,13,1,0,0,0,0,0,0,0,9,26,9,0,0,0,0,
%U 0,0,0,0,4,32,32,4,0,0,0,0,0,0,0,0,1,26,63,26,1,0,0,0,0,0,0,0,0,0,14,80,80,14,0,0,0,0,0
%N Square array, read by antidiagonals, where A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).
%C It appears that the main diagonal (1,1,2,5,11,...) is A051286 (Whitney number of level n of the lattice of the ideals of the fence of size 2 n) that the diagonals (0,1,2,5,13,...) adjacent to the main diagonal are A110320 (Number of blocks in all RNA secondary structures with n nodes) and that the n-th antidiagonal sum = A094686(n-1) (a Fibonacci convolution). The n-th row sum = A002605(n).
%F A(1,1) = A(2,2) = 1, A(1,2) = A(2,1) = 0, A(n,k) = 0 if n < 1 or k < 1, otherwise A(n,k) = A(n-2,k-2) + A(n-1,k-2) + A(n-2,k-1) + A(n-1,k-1).
%F From _Peter Bala_, Nov 07 2017: (Start)
%F T(n,k) = Sum_{i = floor((n+1)/2)..k} binomial(i,n-i)* binomial(i,k-i).
%F Square array = A026729 * transpose(A026729), where A026729 is viewed as a lower unit triangular array. Omitting the first row and column of square array = A030528 * transpose(A030528).
%F O.g.f. 1/(1 - t*(1 + t)*x - t*(1 + t)*x^2) = 1 + (t + t^2)*x + (t + 2*t^2 + 2*t^3 + t^4)*x^2 + .... Cf. A109466 with o.g.f. 1/(1 - t*x - t*x^2).
%F The n-th row polynomial R(n,t) satisfies R(n,t) = R(n,-1 - t).
%F R(n,t) = (-1)^n*sqrt(-t*(1 + t))^n*U(n, 1/2*sqrt(-t*(1 + t))), where U(n,x) denotes the n-th Chebyshev polynomial of the second kind.
%F The sequence of row polynomials R(n,t) is a divisibility sequence of polynomials, that is, if m divides n then R(m,t) divides R(n,t) in the polynomial ring Z[t].
%F R(n,1) = A002605; R(n,2) = A057089. (End)
%e Array starts as:
%e 1 0 0 0 0 0 0 ...
%e 0 1 1 0 0 0 0 ...
%e 0 1 2 2 1 0 0 ...
%e 0 0 2 5 5 3 1 0 ...
%e 0 0 1 5 11 13 9 4 1 0...
%e 0 0 0 3 13 26 32 26 14 5 1 0 ...
%e 0 0 0 1 9 32 63 80 71 45 20 6 1 0 ...
%e 0 0 0 0 4 26 80 153 201 191 135 71 27 7 1 0 ...
%e ...
%t T[n_, k_] := Sum[Binomial[i, n-i] Binomial[i, k-i], {i, Floor[(n+1)/2], k}];
%t Table[T[n-k, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 12 2019 *)
%o (PARI) A=matrix(22,22);A[1,1]=1;A[2,2]=1;A[2,1]=0;A[1,2]=0;A[3,2]=1;A[2,3]=1; for(n=3,22,for(k=3,22,A[n,k]=A[n-2,k-2]+A[n-1,k-2]+A[n-2,k-1]+A[n-1,k-1])); for(n=1,22,for(i=1,n,print1(A[n-i+1,i],", ")))
%Y Cf. A051286, A110320, A002605, A026729, A030528, A057089, A109466.
%K nonn,tabl,easy
%O 1,13
%A _Gerald McGarvey_, Jan 15 2007
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