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A135226 Triangle A135225 + A007318 - A103451, read by rows.

%I A135226 #15 Mar 27 2022 19:03:07
%S A135226 1,1,1,1,3,1,1,4,5,1,1,5,9,7,1,1,6,14,16,9,1,1,7,20,30,25,11,1,1,8,27,
%T A135226 50,55,36,13,1,1,9,35,77,105,91,49,15,1,1,10,44,112,182,196,140,64,17,
%U A135226 1,1,11,54,156,294,378,336,204,81,19,1
%N A135226 Triangle A135225 + A007318 - A103451, read by rows.
%C A135226 Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).
%H A135226 G. C. Greubel, <a href="/A135226/b135226.txt">Rows n = 0..100 of triangle, flattened</a>
%F A135226 T(n,k) = A135225(n,k) + A007318(n,k) - A103451(n,k) as infinite lower triangular matrices.
%F A135226 T(n,k) = ((n+k)/n)*binomial(n,k) with T(n,0) = T(n,n) = 1. - _G. C. Greubel_, Nov 20 2019
%e A135226 First few rows of the triangle:
%e A135226   1;
%e A135226   1, 1;
%e A135226   1, 3,  1;
%e A135226   1, 4,  5,  1;
%e A135226   1, 5,  9,  7,  1;
%e A135226   1, 6, 14, 16,  9,  1;
%e A135226   1, 7, 20, 30, 25, 11,  1;
%e A135226   1, 8, 27, 50, 55, 36, 13, 1;
%e A135226 ...
%p A135226 T:= proc(n, k) option remember;
%p A135226       if k=0 or k=n then 1
%p A135226     else ((n+k)/n)*binomial(n,k)
%p A135226       fi; end:
%p A135226 seq(seq(T(n, k), k=0..n), n=0..10); # _G. C. Greubel_, Nov 20 2019
%t A135226 T[n_, k_]:= T[n, k]= If[k==0 || k==n, 1, ((n+k)/n) Binomial[n, k]]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o A135226 (PARI) T(n,k) = if(k==0 || k==n, 1, ((n+k)/n)*binomial(n,k)); \\ _G. C. Greubel_, Nov 20 2019
%o A135226 (Magma)
%o A135226 T:= func< n,k | k eq 0 or k eq n select 1 else ((n+k)/n)*Binomial(n,k) >;
%o A135226 [T(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 20 2019
%o A135226 (Sage)
%o A135226 @CachedFunction
%o A135226 def T(n,k):
%o A135226     if (k==0 or k==n): return 1
%o A135226     else: return ((n+k)/n)*binomial(n, k)
%o A135226 [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 20 2019
%o A135226 (GAP)
%o A135226 T:= function(n,k)
%o A135226     if k=0 or k=n then return 1;
%o A135226     else return ((n+k)/n)*Binomial(n,k);
%o A135226     fi; end;
%o A135226 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 20 2019
%Y A135226 Cf. A007318, A083329, A103451, A135225.
%K A135226 nonn,tabl
%O A135226 0,5
%A A135226 _Gary W. Adamson_, Nov 23 2007
%E A135226 Corrected and extended by _Philippe Deléham_, Nov 14 2011