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

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, 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, 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

Offset: 0

Author

Reinhard Zumkeller, Mar 04 2010

Keywords

• nonn
• tabl

Comments

T(n,k) = SUM(bn(i)*bk(L-i-1): 0<=iA070939(n), n=SUM(bn(i)*2^i:0<=i

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

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

row sums give A173921; central terms give A159780;

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

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

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

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

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

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

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

Examples

			T(13,10) = T('1101','1010') = 1*0 + 1*1 + 0*0 + 1*1 = 2;
T(13,11) = T('1101','1011') = 1*1 + 1*1 + 0*0 + 1*1 = 3;
T(13,12) = T('1101','1100') = 1*0 + 1*0 + 0*1 + 1*1 = 1;
T(13,13) = T('1101','1101') = 1*1 + 1*0 + 0*1 + 1*1 = 2.
Triangle begins:
  0;
  0, 1;
  0, 1, 0;
  0, 1, 1, 2;
  0, 1, 0, 1, 0;
  0, 1, 0, 1, 1, 2;
  ...
		

Links

• R. Zumkeller, Rows 0 to 320 of the triangle, flattened
• Index entries for sequences related to binary expansion of n

Programs

Mathematica

T[n_, k_] := Module[{bn, bk, lg},
     bn = IntegerDigits[n, 2];
     bk = IntegerDigits[k, 2];
     lg = Max[Length[bn], Length[bk]];
     ListConvolve[PadLeft[bn, lg], PadLeft[bk, lg]]][[1]];
Table[T[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

Formula

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)).

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