Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #15 Sep 05 2020 20:08:37
%S 1,2,3,2,2,4,7,2,2,4,6,4,4,8,15,2,2,4,6,4,4,8,14,4,4,8,12,8,8,16,31,2,
%T 2,4,6,4,4,8,14,4,4,8,12,8,8,16,30,4,4,8,12,8,8,16,28,8,8,16,24,16,16,
%U 32,63,2,2,4,6,4,4,8,14,4,4,8,12,8,8,16,30,4,4,8
%N a(n) = number of odd numbers in the n-th row of the Narayana triangle A001263.
%C a(n)=n iff n=2^k-1 or n=2.
%H S.P. Eu, S.C. Liu, Y.N. Yeh, <a href="https://doi.org/10.1111/j.1467-9590.2006.00337.x">On the Congruences of Some Combinatorial Numbers</a>, Studies in Applied Mathematics, 116(2006), 135-144.
%e The Narayana numbers are binomial(n-1, k-1)*binomial(n, k-1)/k. a(4)=4 since for n=4 there are two odd numbers among 1,6,6,1.
%t a[n_] := Count[Table[Binomial[n - 1, k - 1] Binomial[n, k - 1]/k, {k, 1, n}], _?OddQ]; Array[a, 100] (* _Amiram Eldar_, Jul 02 2020 *)
%o (PARI) a(n) = sum(k=1, n, binomial(n-1, k-1)*binomial(n, k-1)/k % 2); \\ _Michel Marcus_, Jul 02 2020
%Y Cf. A001263.
%K nonn
%O 1,2
%A _Sen-Peng Eu_, Jul 01 2020