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A135228 Triangle A000012(signed) * A007318 * A103451, read by rows.

%I A135228 #9 Sep 08 2022 08:45:32
%S A135228 1,1,1,3,1,1,5,2,2,1,11,2,4,3,1,21,3,6,7,4,1,43,3,9,13,11,5,1,85,4,12,
%T A135228 22,24,16,6,1,171,4,16,34,46,40,22,7,1,341,5,20,50,80,86,62,29,8,1,
%U A135228 683,5,25,70,130,166,148,91,37,9,1,1365,6,30,95,200,296,314,239,128,46,10,1
%N A135228 Triangle A000012(signed) * A007318 * A103451, read by rows.
%C A135228 Row sums = A000975: (1, 2, 5, 10, 21, 42, 85, 170, ...).
%C A135228 Left border = A001045: (1, 1, 3, 5, 11, 21, 43, ...).
%H A135228 G. C. Greubel, <a href="/A135228/b135228.txt">Rows n = 0..100 of triangle, flattened</a>
%F A135228 T(n,k) = A000012(signed) * A007318 * A103451 as infinite lower triangular matrices. A000012(signed) = (1; -1,1; 1,-1,1; ...).
%F A135228 T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) =
%F A135228 (2^(n+1) - (-1)^(n+1))/3 (Jacobsthal_{n+1}).- _G. C. Greubel_, Nov 20 2019
%e A135228 First few rows of the triangle are:
%e A135228     1;
%e A135228     1, 1;
%e A135228     3, 1,  1;
%e A135228     5, 2,  2,  1;
%e A135228    11, 2,  4,  3,  1;
%e A135228    21, 3,  6,  7,  4,  1;
%e A135228    43, 3,  9, 13, 11,  5,  1;
%e A135228    85, 4, 12, 22, 24, 16,  6, 1;
%e A135228   171, 4, 16, 34, 46, 40, 22, 7, 1;
%e A135228 ...
%p A135228 T:= proc(n, k) option remember;
%p A135228       if k=0 then (2^(n+1) +(-1)^n)/3
%p A135228     else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
%p A135228       fi; end:
%p A135228 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019
%t A135228 T[n_, k_]:= T[n, k]= If[k==0, (2^(n+1) +(-1)^n)/3, 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 A135228 (PARI) T(n,k) = if(k==0, (2^(n+1) +(-1)^n)/3, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) ); \\ _G. C. Greubel_, Nov 20 2019
%o A135228 (Magma)
%o A135228 function T(n,k)
%o A135228   if k eq 0 then return (2^(n+1) +(-1)^n)/3;
%o A135228   else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
%o A135228   end if; return T; end function;
%o A135228 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019
%o A135228 (Sage)
%o A135228 @CachedFunction
%o A135228 def T(n, k):
%o A135228     if (k==0): return (2^(n+1) +(-1)^n)/3
%o A135228     else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
%o A135228 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019
%Y A135228 Cf. A000975, A001045, A007318, A103451.
%K A135228 nonn,tabl
%O A135228 0,4
%A A135228 _Gary W. Adamson_, Nov 23 2007
%E A135228 Offset changed by _G. C. Greubel_, Nov 20 2019