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A135229 Triangle A000012(signed) * A103451 * A007318, read by rows.

%I A135229 #11 Sep 08 2022 08:45:32
%S A135229 1,1,1,1,1,1,1,2,2,1,1,2,4,3,1,1,3,6,7,4,1,1,3,9,13,11,5,1,1,4,12,22,
%T A135229 24,16,6,1,1,4,16,34,46,40,22,7,1,1,5,20,50,80,86,62,29,8,1
%N A135229 Triangle A000012(signed) * A103451 * A007318, read by rows.
%C A135229 row sums = A005578 starting (1, 2, 3, 6, 11, 22, 43, 86, ...).
%H A135229 G. C. Greubel, <a href="/A135229/b135229.txt">Rows n = 0..100 of triangle, flattened</a>
%F A135229 T(n,k) = A000012(signed) * A103451 * A007318 as infinite lower triangular matrices, where A000012(signed) = (1; -1,1; 1,-1,1; ...).
%F A135229 T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = 1. - _G. C. Greubel_, Nov 20 2019
%e A135229 First few rows of the triangle are:
%e A135229   1;
%e A135229   1, 1;
%e A135229   1, 1,  1;
%e A135229   1, 2,  2,  1;
%e A135229   1, 2,  4,  3,  1;
%e A135229   1, 3,  6,  7,  4,  1;
%e A135229   1, 3,  9, 13, 11,  5,  1;
%e A135229   1, 4, 12, 22, 24, 16,  6, 1;
%e A135229   1, 4, 16, 34, 46, 40, 22, 7, 1;
%e A135229 ...
%p A135229 T:= proc(n, k) option remember;
%p A135229       if k=0 then 1
%p A135229     else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
%p A135229       fi; end:
%p A135229 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019
%t A135229 T[n_, k_]:= T[n, k]= If[k==0, 1, Sum[Binomial[n-1-2*j, k-1], {j, 0, Floor[(n-1)/2]}]]; Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 20 2019 *)
%o A135229 (PARI) T(n,k) = if(k==0, 1, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ _G. C. Greubel_, Nov 20 2019
%o A135229 (Magma)
%o A135229 function T(n,k)
%o A135229   if k eq 0 then return 1;
%o A135229   else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
%o A135229   end if; return T; end function;
%o A135229 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019
%o A135229 (Sage)
%o A135229 @CachedFunction
%o A135229 def T(n, k):
%o A135229     if (k==0): 1
%o A135229     else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
%o A135229 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019
%Y A135229 Cf. A005578, A007318, A103451.
%K A135229 nonn,tabl
%O A135229 0,8
%A A135229 _Gary W. Adamson_, Nov 23 2007
%E A135229 Offset changed by _G. C. Greubel_, Nov 20 2019