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

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

Offset

0,2

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

Links

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

Formula

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

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A153435 A266787 A109227 * A110574 A280042 A287044

Adjacent sequences: A133339 A133340 A133341 * A133343 A133344 A133345

Keyword

base,easy,nonn

Author

Jonathan Vos Post, Oct 20 2007

Extensions

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

Status

approved