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A135230 Triangle A103451 * A000012(signed) * A007318, read by rows.

%I A135230 #7 Sep 08 2022 08:45:32
%S A135230 1,1,1,2,1,1,1,2,2,1,2,2,4,3,1,1,3,6,7,4,1,2,3,9,13,11,5,1,1,4,12,22,
%T A135230 24,16,6,1,2,4,16,34,46,40,22,7,1,1,5,20,50,80,86,62,29,8,1,2,5,25,70,
%U A135230 130,166,148,91,37,9,1,1,6,30,95,200,296,314,239,128,46,10,1
%N A135230 Triangle A103451 * A000012(signed) * A007318, read by rows.
%C A135230 row sums = A135231
%H A135230 G. C. Greubel, <a href="/A135230/b135230.txt">Rows n = 0..100 of triangle, flattened</a>
%F A135230 T(n,k) = A103451 * A000012(signed) * A007318, where A000012(signed) = (1; -1,1; 1,-1,1;...).
%F A135230 T(n,k) = Sum_{j=0..floor((n-1)/2)} binomial(n-2*j-1, k-1), with T(n,0) = (3+(-1)^n)/2 and T(n,n) = 1. - _G. C. Greubel_, Nov 20 2019
%e A135230 First few rows of the triangle are:
%e A135230   1;
%e A135230   1, 1;
%e A135230   2, 1,  1;
%e A135230   1, 2,  2,  1;
%e A135230   2, 2,  4,  3,  1;
%e A135230   1, 3,  6,  7,  4,  1;
%e A135230   2, 3,  9, 13, 11,  5,  1;
%e A135230   1, 4, 12, 22, 24, 16,  6, 1;
%e A135230   2, 4, 16, 34, 46, 40, 22, 7, 1;
%e A135230 ...
%p A135230 T:= proc(n, k) option remember;
%p A135230       if k=n then 1
%p A135230     elif k=0 then (3+(-1)^n)/2
%p A135230     else add(binomial(n-2*j-1, k-1), j=0..floor((n-1)/2))
%p A135230       fi; end:
%p A135230 seq(seq(T(n, k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 20 2019
%t A135230 T[n_, k_]:= T[n, k]= If[k==n, 1, If[k==0, (3+(-1)^n)/2, 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 A135230 (PARI) T(n,k) = if(k==n, 1, if(k==0, (3+(-1)^n)/2, sum(j=0, (n-1)\2, binomial( n-2*j-1, k-1)) )); \\ _G. C. Greubel_, Nov 20 2019
%o A135230 (Magma)
%o A135230 function T(n,k)
%o A135230   if k eq n then return 1;
%o A135230   elif k eq 0 then return (3+(-1)^n)/2;
%o A135230   else return (&+[Binomial(n-2*j-1, k-1): j in [0..Floor((n-1)/2)]]);
%o A135230   end if; return T; end function;
%o A135230 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2019
%o A135230 (Sage)
%o A135230 @CachedFunction
%o A135230 def T(n, k):
%o A135230     if (k==n): return 1
%o A135230     elif (k==0): return (3+(-1)^n)/2
%o A135230     else: return sum(binomial(n-2*j-1, k-1) for j in (0..floor((n-1)/2)))
%o A135230 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 20 2019
%Y A135230 Cf. A000012, A007318, A103451, A135231.
%K A135230 nonn,tabl
%O A135230 0,4
%A A135230 _Gary W. Adamson_, Nov 23 2007
%E A135230 More terms and offset changed by _G. C. Greubel_, Nov 20 2019