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A173920 Triangle read by rows: T(n,k) = convolution of n with k in binary representation, 0<=k<=n.

%I A173920 #10 Sep 19 2021 12:10:34
%S A173920 0,0,1,0,1,0,0,1,1,2,0,1,0,1,0,0,1,0,1,1,2,0,1,1,2,0,1,1,0,1,1,2,1,2,
%T A173920 2,3,0,1,0,1,0,1,0,1,0,0,1,0,1,0,1,0,1,1,2,0,1,0,1,1,2,1,2,0,1,0,0,1,
%U A173920 0,1,1,2,1,2,1,2,1,2,0,1,1,2,0,1,1,2,0,1,1,2,0,0,1,1,2,0,1,1,2,1,2,2,3,1,2
%N A173920 Triangle read by rows: T(n,k) = convolution of n with k in binary representation, 0<=k<=n.
%C A173920 T(n,k) = SUM(bn(i)*bk(L-i-1): 0<=i<L), where L=A070939(n), n=SUM(bn(i)*2^i:0<=i<L), and k=SUM(bk(i)*2^i:0<=i<L);
%C A173920 T(n,2*k+1) = T(n,2*k) + 1;
%C A173920 T(n,k) <= MIN{A000120(n),A000120(k)};
%C A173920 row sums give A173921; central terms give A159780;
%C A173920 T(n,0) = A000004(n);
%C A173920 T(n,1) = A000012(n) for n>0;
%C A173920 T(n,2) = A079944(n-2) for n>1;
%C A173920 T(n,3) = A079882(n-2) for n>2;
%C A173920 T(n,4) = A173922(n-4) for n>3;
%C A173920 T(n,8) = A173923(n-8) for n>7;
%C A173920 T(n,n) = A159780(n).
%H A173920 R. Zumkeller, <a href="/A173920/b173920.txt">Rows 0 to 320 of the triangle, flattened</a>
%H A173920 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A173920 T(n,k) = c(A030101(n),k,0) with c(x,y,z) = if y=0 then z else c([x/2],[y/2],z+(x mod 2)*(y mod 2)).
%e A173920 T(13,10) = T('1101','1010') = 1*0 + 1*1 + 0*0 + 1*1 = 2;
%e A173920 T(13,11) = T('1101','1011') = 1*1 + 1*1 + 0*0 + 1*1 = 3;
%e A173920 T(13,12) = T('1101','1100') = 1*0 + 1*0 + 0*1 + 1*1 = 1;
%e A173920 T(13,13) = T('1101','1101') = 1*1 + 1*0 + 0*1 + 1*1 = 2.
%e A173920 Triangle begins:
%e A173920   0;
%e A173920   0, 1;
%e A173920   0, 1, 0;
%e A173920   0, 1, 1, 2;
%e A173920   0, 1, 0, 1, 0;
%e A173920   0, 1, 0, 1, 1, 2;
%e A173920   ...
%t A173920 T[n_, k_] := Module[{bn, bk, lg},
%t A173920      bn = IntegerDigits[n, 2];
%t A173920      bk = IntegerDigits[k, 2];
%t A173920      lg = Max[Length[bn], Length[bk]];
%t A173920      ListConvolve[PadLeft[bn, lg], PadLeft[bk, lg]]][[1]];
%t A173920 Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2021 *)
%K A173920 nonn,tabl
%O A173920 0,10
%A A173920 _Reinhard Zumkeller_, Mar 04 2010