A363882 Take 2 copies of Pascal's triangle. One copy has one inch between the terms of each row and the other copy has two inches between the terms of each row. Put one on top of the other so that the 1's at the very top of each copy coincide. Sequence is a triangle giving the differences between the overlapping terms.

0, 1, 1, 0, 2, 0, 1, 3, 3, 1, 0, 4, 4, 4, 0, 1, 5, 10, 10, 5, 1, 0, 6, 12, 20, 12, 6, 0, 1, 7, 21, 35, 35, 21, 7, 1, 0, 8, 24, 56, 64, 56, 24, 8, 0, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 0, 10, 40, 120, 200, 252, 200, 120, 40, 10, 0, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset

0,5

Links

Table of n, a(n) for n=0..77.

Formula

T(n,k) = binomial(n,k) - [n mod 2 = k mod 2 = 0] * binomial(n/2,k/2).

Example

Row n=8 is formed by taking row 8 of Pascal's triangle (1, 8, 28, 56, 70, 56, 28, 8, 1) and subtracting row 4 (1, 4, 6, 4, 1) spaced 2 apart. The numbers that overlap are 1, 28, 70, 28, 1 over 1, 4, 6, 4, 1, from which 1-1=0, 28-4=24, 70-6=64, 28-4=24, and 1-1=0. Thus, row 8 of the present triangle is 0, 8, 24, 56, 64, 56, 24, 8, 0.
Subtracting:
  1;                1;               0,
  1, 1;                              1, 1;
  1, 2, 1;       -  1,    1;      =  0, 2, 0;
  1, 3, 3, 1;                        1, 3, 3, 1;
  1, 4, 6, 4, 1;    1,    2,    1;   0, 4, 4, 4, 0;
  ...
Resulting triangle begins:
      k=0  1  2  3  4
  n=0:  0;
  n=1:  1, 1;
  n=2:  0, 2, 0;
  n=3:  1, 3, 3, 1;
  n=4:  0, 4, 4, 4, 0;
  ...

Crossrefs

Cf. A007318.

Sequence in context: A323073 A167279 A068920 * A099390 A297477 A370030

Adjacent sequences: A363879 A363880 A363881 * A363883 A363884 A363885

Keyword

tabl,nonn,easy

Author

J. Lowell, Jun 25 2023

Status

approved