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A034868 Left half of Pascal's triangle.

%I A034868 #45 Oct 18 2023 10:03:19
%S A034868 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,
%T A034868 126,1,10,45,120,210,252,1,11,55,165,330,462,1,12,66,220,495,792,924,
%U A034868 1,13,78,286,715,1287,1716,1,14,91,364,1001,2002,3003,3432,1,15
%N A034868 Left half of Pascal's triangle.
%H A034868 Reinhard Zumkeller, <a href="/A034868/b034868.txt">Rows n=0..150 of triangle, flattened</a>
%H A034868 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A034868 T(n,k) = A034869(n,floor(n/2)-k), k = 0..floor(n/2). - _Reinhard Zumkeller_, Jul 27 2012
%e A034868 1;
%e A034868 1;
%e A034868 1, 2;
%e A034868 1, 3;
%e A034868 1, 4,  6;
%e A034868 1, 5, 10;
%e A034868 1, 6, 15, 20;
%e A034868 ...
%t A034868 Flatten[ Table[ Binomial[n, k], {n, 0, 15}, {k, 0, Floor[n/2]}]] (* _Robert G. Wilson v_, May 28 2005 *)
%o A034868 (Haskell)
%o A034868 a034868 n k = a034868_tabf !! n !! k
%o A034868 a034868_row n = a034868_tabf !! n
%o A034868 a034868_tabf = map reverse a034869_tabf
%o A034868 -- _Reinhard Zumkeller_, improved Dec 20 2015, Jul 27 2012
%o A034868 (PARI) for(n=0, 14, for(k=0, floor(n/2), print1(binomial(n, k),", ");); print();) \\ _Indranil Ghosh_, Mar 31 2017
%o A034868 (Python)
%o A034868 import math
%o A034868 from sympy import binomial
%o A034868 for n in range(15):
%o A034868     print([binomial(n, k) for k in range(int(math.floor(n/2)) + 1)]) # _Indranil Ghosh_, Mar 31 2017
%o A034868 (Python)
%o A034868 from itertools import count, islice
%o A034868 def A034868_gen(): # generator of terms
%o A034868     yield from (s:=(1,))
%o A034868     for i in count(0):
%o A034868         yield from (s:=(1,)+tuple(s[j]+s[j+1] for j in range(len(s)-1)) + ((s[-1]<<1,) if i&1 else ()))
%o A034868 A034868_list = list(islice(A034868_gen(),30)) # _Chai Wah Wu_, Oct 17 2023
%Y A034868 Cf. A007318, A107430, A062344, A122366, A027306 (row sums).
%Y A034868 Cf. A008619.
%Y A034868 Cf. A225860.
%Y A034868 Cf. A126257.
%Y A034868 Cf. A034869 (right half), A014413, A014462, A265848.
%K A034868 nonn,tabf,easy
%O A034868 0,4
%A A034868 _N. J. A. Sloane_