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A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.

%I A109447 #21 Feb 08 2014 00:32:04
%S A109447 1,2,1,3,4,4,1,10,5,6,20,6,1,21,35,7,8,56,56,8,1,36,126,84,9,10,120,
%T A109447 252,120,10,1,55,330,462,165,11,12,220,792,792,220,12,1,78,715,1716,
%U A109447 1287,286,13,14,364,2002,3432,2002,364,14,1,105,1365,5005,6435,3003,455,15
%N A109447 Binomial coefficients C(n,k) with n-k odd, read by rows.
%C A109447 The same as A119900 without 0's. A reflected version of A034867 or A202064. - _Alois P. Heinz_, Feb 07 2014
%C A109447 From _Vladimir Shevelev_, Feb 07 2014: (Start)
%C A109447 Also table of coefficients of polynomials  P_1(x)=1, P_2(x)=2, for n>=2, P_(n+1)(x) = 2*P_n(x)+(x-1)* P_(n-1)(x).  The polynomials P_n(x)/2^(n-1) are connected with sequences A000045 (x=5), A001045 (x=9), A006130 (x=13), A006131 (x=17), A015440 (x=21), A015441 (x=25), A015442 (x=29), A015443 (x=33), A015445 (x=37), A015446 (x=41), A015447 (x=45), A053404 (x=49); also the polynomials P_n(x) are connected with sequences A000129, A002605, A015518, A063727, A085449, A002532, A083099, A015519, A003683, A002534, A083102, A015520. (End)
%H A109447 Alois P. Heinz, <a href="/A109447/b109447.txt">Rows n = 1..200, flattened</a>
%e A109447 Starred terms in Pascal's triangle (A007318), read by rows:
%e A109447 1;
%e A109447 1*, 1;
%e A109447 1, 2*, 1;
%e A109447 1*, 3, 3*, 1;
%e A109447 1, 4*, 6, 4*, 1;
%e A109447 1*, 5, 10*, 10, 5*, 1;
%e A109447 1, 6*, 15, 20*, 15, 6*, 1;
%e A109447 1*, 7, 21*, 35, 35*, 21, 7*, 1;
%e A109447 1, 8*, 28, 56*, 70, 56*, 28, 8*, 1;
%e A109447 1*, 9, 36*, 84, 126*, 126, 84*, 36, 9*, 1;
%e A109447 Triangle T(n,k) begins:
%e A109447 1;
%e A109447 2;
%e A109447 1,    3;
%e A109447 4,    4;
%e A109447 1,   10,  5;
%e A109447 6,   20,  6;
%e A109447 1,   21,  35,   7;
%e A109447 8,   56,  56,   8;
%e A109447 1,   36, 126,  84,  9;
%e A109447 10, 120, 252, 120, 10;
%p A109447 T:= (n, k)-> binomial(n, 2*k+1-irem(n, 2)):
%p A109447 seq(seq(T(n, k), k=0..ceil((n-2)/2)), n=1..20);  # _Alois P. Heinz_, Feb 07 2014
%t A109447 Flatten[ Table[ If[ OddQ[n - k], Binomial[n, k], {}], {n, 0, 15}, {k, 0, n}]] (* _Robert G. Wilson v_ *)
%Y A109447 Cf. A109446.
%K A109447 easy,nonn,tabf
%O A109447 1,2
%A A109447 _Philippe Deléham_, Aug 27 2005
%E A109447 More terms from _Robert G. Wilson v_, Aug 30 2005
%E A109447 Corrected offset by _Alois P. Heinz_, Feb 07 2014