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%I #14 Mar 27 2022 19:01:55
%S 1,3,1,5,3,1,9,7,4,1,17,15,11,5,1,33,31,26,16,6,1,65,63,57,42,22,7,1,
%T 129,127,120,99,64,29,8,1,257,255,247,219,163,93,37,9,1,513,511,502,
%U 466,382,256,130,46,10,1
%N Triangle A103451 * A007318 * A000012, read by rows. T(n, k) for 0 <= k <= n.
%C Row sums = A132750: (1, 4, 9, 21, 49, 113, ...).
%C Left border = A083318: (1, 3, 5, 9, 17, 33, ...).
%H G. C. Greubel, <a href="/A135224/b135224.txt">Rows n = 0..100 of triangle, flattened</a>
%F T(n, k) = A103451(n,k) * A007318(n,k) * A000012(n,k) as infinite lower triangular matrices.
%F T(n, k) = Sum_{j=0..n} binomial(n, k+j), with T(0,0) = 1 and T(n,0) = 2^n + 1. - _G. C. Greubel_, Nov 20 2019
%F T(n, k) = binomial(n, k)*hypergeom([1, k-n], [k+1], -1) - binomial(n, k+n+1)* hypergeom([1, k+1], [k+n+2], -1) + 0^k - 0^n. - _Peter Luschny_, Nov 20 2019
%e First few rows of the triangle:
%e 1;
%e 3, 1;
%e 5, 3, 1;
%e 9, 7, 4, 1;
%e 17, 15, 11, 5, 1;
%e 33, 31, 26, 16, 6, 1;
%e 65, 63, 57, 42, 22, 7, 1;
%e ...
%p T:= proc(n, k) option remember;
%p if k=0 and n=0 then 1
%p elif k=0 then 2^n +1
%p else add(binomial(n, k+j), j=0..n)
%p fi; end:
%p seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 20 2019
%t T[n_, k_]:= T[n, k] = If[k==n==0, 1, If[k==0, 2^n +1, Sum[Binomial[n, k + j], {j, 0, n}]]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o (PARI) T(n,k) = if(k==0 && n==0, 1, if(k==0, 2^n +1, sum(j=0, n, binomial(n, k+j)) )); \\ _G. C. Greubel_, Nov 20 2019
%o (Magma)
%o function T(n,k)
%o if k eq 0 and n eq 0 then return 1;
%o elif k eq 0 then return 2^n +1;
%o else return (&+[Binomial(n, k+j): j in [0..n]]);
%o end if; return T; end function;
%o [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 20 2019
%o (Sage)
%o def T(n, k):
%o if (k==0 and n==0): return 1
%o elif (k==0): return 2^n + 1
%o else: return sum(binomial(n, k+j) for j in (0..n))
%o [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 20 2019
%Y Cf. A083318, A103451, A132750.
%K nonn,tabl
%O 0,2
%A _Gary W. Adamson_, Nov 23 2007
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