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%I #28 Aug 07 2020 12:10:56
%S 1,2,2,4,6,4,8,14,18,8,16,30,48,56,16,32,62,110,166,182,32,64,126,236,
%T 402,584,616,64,128,254,490,892,1476,2092,2156,128,256,510,1000,1892,
%U 3368,5460,7616,7744,256,512,1022,2022,3914,7282,12742,20358,28102,28358,512
%N Triangle T(n,k), read by rows, formed by setting all entries in the zeroth column ((n,0) entries) and the main diagonal ((n,n) entries) to powers of 2 with all other entries formed by the recursion T(n,k) = T(n-1,k) + T(n,k-1).
%C T(n,k) = T(n-1,k) + T(n,k-1) for n >= 2 and 1 <= k <= n - 1 with T(n,0) = T(n,n) = 2^n for n >= 0.
%C The n-th row sum equals A082590(n), which is the expansion of 1/(1 - 2*x)/sqrt(1 - 4*x) and equals 2^n * JacobiP(n, 1/2, -1-n, 3).
%C First column is T(n,1) = A000918(n+1) = 2^(n+1) - 2.
%C From _Petros Hadjicostas_, Aug 06 2020: (Start)
%C T(n,2) = 2^(n+2) - 2*n - 8 for n >= 2.
%C T(n+1,n) = 2^n + Sum_{k=0..n} T(n,k) = 2^n + A082590(n).
%C Bivariate o.g.f.: ((1 - x)*(1 - y)/(1 - 2*x) - x*y/sqrt(1 - 4*x*y))/((1 - 2*x*y)*(1 - x - y)). (End)
%e From _Petros Hadjicostas_, Aug 06 2020: (Start)
%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e 1;
%e 2, 2;
%e 4, 6, 4;
%e 8, 14, 18, 8;
%e 16, 30, 48, 56, 16;
%e 32, 62, 110, 166, 182, 32;
%e 64, 126, 236, 402, 584, 616, 64;
%e ... (End)
%o (PARI) T(n,k) = if ((k==0) || (n==k), 2^n, if ((n<0) || (k<0), 0, if (n>k, T(n-1,k) + T(n,k-1), 0)));
%o for(n=0, 10, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Aug 07 2020
%Y Cf. A000918, A082590.
%K nonn,tabl
%O 0,2
%A _Gerald McGarvey_, Aug 12 2004
%E Offset changed to 0 by _Petros Hadjicostas_, Aug 06 2020
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