A173920 - OEIS

%I #10 Sep 19 2021 12:10:34

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

%C 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 T(n,2*k+1) = T(n,2*k) + 1;

%C T(n,k) <= MIN{A000120(n),A000120(k)};

%C row sums give A173921; central terms give A159780;

%C T(n,0) = A000004(n);

%C T(n,1) = A000012(n) for n>0;

%C T(n,2) = A079944(n-2) for n>1;

%C T(n,3) = A079882(n-2) for n>2;

%C T(n,4) = A173922(n-4) for n>3;

%C T(n,8) = A173923(n-8) for n>7;

%C T(n,n) = A159780(n).

%H R. Zumkeller, <a href="/A173920/b173920.txt">Rows 0 to 320 of the triangle, flattened</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F 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 T(13,10) = T('1101','1010') = 1*0 + 1*1 + 0*0 + 1*1 = 2;

%e T(13,11) = T('1101','1011') = 1*1 + 1*1 + 0*0 + 1*1 = 3;

%e T(13,12) = T('1101','1100') = 1*0 + 1*0 + 0*1 + 1*1 = 1;

%e T(13,13) = T('1101','1101') = 1*1 + 1*0 + 0*1 + 1*1 = 2.

%e Triangle begins:

%e   0;

%e   0, 1;

%e   0, 1, 0;

%e   0, 1, 1, 2;

%e   0, 1, 0, 1, 0;

%e   0, 1, 0, 1, 1, 2;

%e   ...

%t T[n_, k_] := Module[{bn, bk, lg},

%t      bn = IntegerDigits[n, 2];

%t      bk = IntegerDigits[k, 2];

%t      lg = Max[Length[bn], Length[bk]];

%t      ListConvolve[PadLeft[bn, lg], PadLeft[bk, lg]]][[1]];

%t Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Sep 19 2021 *)

%K nonn,tabl

%O 0,10

%A _Reinhard Zumkeller_, Mar 04 2010