A034868 Left half of Pascal's triangle.

1, 1, 1, 2, 1, 3, 1, 4, 6, 1, 5, 10, 1, 6, 15, 20, 1, 7, 21, 35, 1, 8, 28, 56, 70, 1, 9, 36, 84, 126, 1, 10, 45, 120, 210, 252, 1, 11, 55, 165, 330, 462, 1, 12, 66, 220, 495, 792, 924, 1, 13, 78, 286, 715, 1287, 1716, 1, 14, 91, 364, 1001, 2002, 3003, 3432, 1, 15

Offset

0,4

Links

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Index entries for triangles and arrays related to Pascal's triangle

Formula

T(n,k) = A034869(n,floor(n/2)-k), k = 0..floor(n/2). - Reinhard Zumkeller, Jul 27 2012

Example

Triangle begins:
  1;
  1;
  1, 2;
  1, 3;
  1, 4,  6;
  1, 5, 10;
  1, 6, 15, 20;
  ...

Mathematica

Flatten[ Table[ Binomial[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}]] (* Robert G. Wilson v, May 28 2005 *)

Prog

(Haskell)
a034868 n k = a034868_tabf !! n !! k
a034868_row n = a034868_tabf !! n
a034868_tabf = map reverse a034869_tabf
-- Reinhard Zumkeller, improved Dec 20 2015, Jul 27 2012
(PARI) for(n=0, 14, for(k=0, floor(n/2), print1(binomial(n, k), ", "); ); print(); ) \\ Indranil Ghosh, Mar 31 2017
(Python)
from sympy import binomial
for n in range(15):
    print([binomial(n, k) for k in range(n//2 + 1)]) # Indranil Ghosh, Mar 31 2017
(Python)
from itertools import count, islice
def A034868_gen(): # generator of terms
    yield from (s:=(1, ))
    for i in count(0):
        yield from (s:=(1, )+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + ((s[-1]<<1, ) if i&1 else ()))
A034868_list = list(islice(A034868_gen(), 30)) # Chai Wah Wu, Oct 17 2023

Crossrefs

Cf. A007318, A107430, A062344, A122366, A027306 (row sums).

Cf. A008619.

Cf. A225860.

Cf. A126257.

Cf. A034869 (right half), A014413, A014462, A265848.

Sequence in context: A295885 A329035 A082904 * A050382 A197956 A054072

Adjacent sequences: A034865 A034866 A034867 * A034869 A034870 A034871

Keyword

nonn,tabf,easy

Author

N. J. A. Sloane

Status

approved