A133342 Concatenation of binary expansion of n-th row of Pascal's triangle.

1, 11, 1101, 111111, 11001101001, 1101101010101011, 111011111010011111101, 111110101100011100011101011111, 110001110011100010001101110001110010001, 11001100100101010011111101111110101010010010010011

Offset: 0

Author

Jonathan Vos Post, Oct 20 2007

Keywords

• base
• easy
• nonn

Comments

Binary analog of A003590. More generally, this sequence is the 2nd row of the matrix whose k-th row is the concatenation of the base-k expression of n-th row of Pascal's triangle. The 10th row of that array is A003590.

a(n) is a repunit for n = 0, 1, 3, but otherwise not since C(n,1) = n has a 0-bit apart from at n=2^k-1 and there C(n,2) = n*(n-1)/2 == 1 (mod 4) has a 0-bit. - Kevin Ryde and Bernard Schott, Nov 11 2021

Examples

			a(0) = 1 because the 0th row of Pascal's triangle is 1.
a(1) = 11 because the 1st row of Pascal's triangle is 1,1 which concatenates to 11.
a(2) = 1101 because the 2nd row of Pascal's triangle is 1,2,1 which in binary is 1,10,1 which concatenates to 1101.
a(3) = 111111 because the 3rd row of Pascal's triangle is 1,3,3,1 which in binary is 1,11,11,1 which concatenates to 111111.
a(4) = 110010101001 because the 4th row of Pascal's triangle is 1,4,6,4,1 which in binary is 1,100,110,100,1 which concatenates to 11001101001.
a(5) = 1101101010101011 because the 5th row of Pascal's triangle is 1,5,10,10,5,1 which in binary is 1,101,1010,1010,101,1 which concatenates to 1101101010101011.
a(6) = 111011111010011111101 because the 6th row of Pascal's triangle is 1,6,15,20,15,6,1 which in binary is 1,110,1111,10100,1111,110,1 which concatenates to 111011111010011111101.
The array of base k concatenations begins:
  k/n  0  1   2    3        4
  1.|  1  11  1111 11111111 1111111111111111  2^(n-1) repetitions of 1
  2.|  1  11  1101 111111   11001101001
  3.|  1  11  121  110101   11120111
  4.|  1  11  121  1331     11012101
  5.|  1  11  121  1331     141141
  6.|  1  11  121  1331     141041
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..38

Crossrefs

Cf. A000217, A002275, A003590, A007318, A349304.

Programs

Maple

catL := proc(L) local resul,a ; resul:=0 ; for a in L do resul := resul*10^(max(ilog10(a)+1,1))+a ; od: RETURN(resul) ; end: A133342 := proc(n) local prow,k ; prow := [1] ; for k from 1 to n do prow := [op(prow), convert(binomial(n,k),binary) ] ; od: catL(prow) ; end: seq(A133342(n),n=0..11) ; # R. J. Mathar, Jan 08 2008

Mathematica

FromDigits[Flatten[IntegerDigits[#,2]]]&/@Table[Binomial[n,k],{n,0,10},{k,0,n}] (* Harvey P. Dale, Apr 12 2020 *)

Formula

a(n) = Concatenate[k=1..n] binomial(n, k) (base 2).

a(n) = Concatenate[i=A000217(n)..A000217(n+1)] A007088(A007318(i)).

Extensions

Corrected and extended by R. J. Mathar, Jan 08 2008