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%I #10 Apr 15 2017 05:29:37
%S 1,1,1,1,1,1,1,2,1,1,3,2,3,1,1,3,9,3,4,1,1,13,9,18,4,5,1,1,13,42,18,
%T 30,5,6,1,1,73,42,95,30,45,6,7,1,1,73,270,95,179,45,63,7,8,1,1,466,
%U 270,693,179,301,63,84,8,9,1,1,466,1785,693,1463,301,468,84,108,9,10,1,1
%N Triangle, read by rows, where T(n,k) = {T^[(n+k)/2]}( [(n+k)/2], k) for n>=k>=0, so that antidiagonal {2n} equals row n of T^n for n>=0 and odd antidiagonals equal even antidiagonals.
%C Column k of triangle A132845 is equal to column k of this triangle but without repetition of terms.
%H G. C. Greubel, <a href="/A132844/b132844.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F T(n,k) = A132845( [(n+k)/2], k) where A132845(n,k) = [T^n](n,k) for n>=k>=0.
%e Triangle T begins:
%e 1;
%e 1, 1;
%e 1, 1, 1;
%e 1, 2, 1, 1;
%e 3, 2, 3, 1, 1;
%e 3, 9, 3, 4, 1, 1;
%e 13, 9, 18, 4, 5, 1, 1;
%e 13, 42, 18, 30, 5, 6, 1, 1;
%e 73, 42, 95, 30, 45, 6, 7, 1, 1;
%e 73, 270, 95, 179, 45, 63, 7, 8, 1, 1;
%e 466, 270, 693, 179, 301, 63, 84, 8, 9, 1, 1;
%e 466, 1785, 693, 1463, 301, 468, 84, 108, 9, 10, 1, 1;
%e 3309, 1785, 4893, 1463, 2726, 468, 687, 108, 135, 10, 11, 1, 1; ...
%e Matrix square T^2 begins:
%e 1;
%e 2, 1;
%e 3, 2, 1; <-- antidiagonals 4, 5, of T
%e 5, 5, 2, 1;
%e 12, 9, 7, 2, 1;
%e 25, 31, 13, 9, 2, 1;
%e 75, 63, 58, 17, 11, 2, 1; ...
%e Matrix cube T^3 begins:
%e 1;
%e 3, 1;
%e 6, 3, 1;
%e 13, 9, 3, 1; <-- antidiagonals 6, 7, of T
%e 33, 22, 12, 3, 1;
%e 87, 75, 31, 15, 3, 1;
%e 265, 204, 132, 40, 18, 3, 1; ...
%e Matrix 4th power T^4 begins:
%e 1;
%e 4, 1;
%e 10, 4, 1;
%e 26, 14, 4, 1;
%e 73, 42, 18, 4, 1; <-- antidiagonals 8, 9, of T
%e 220, 151, 58, 22, 4, 1;
%e 717, 488, 253, 74, 26, 4, 1; ...
%t t[n_, k_] := t[n, k] = Module[{q = Quotient[n+k, 2], m, p}, m = Table[ Which[c < r-1, t[r-1, c-1], c <= r, 1, True, 0], {r, 1, q+1}, {c, 1, q+1}]; p = MatrixPower[m, q]; If[k > q, 0, p[[q+1, k+1]]]]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 02 2013, after Pari *)
%o (PARI) {T(n,k)=local(M=matrix((n+k)\2+1,(n+k)\2+1,r,c,if(r>=c,if(r<=c+1,1,T(r-1,c-1))))); (M^((n+k)\2))[(n+k)\2+1,k+1]}
%Y Cf. A132845 (triangle); columns: A132846, A132847, A132848, A132849.
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Sep 17 2007
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