Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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