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A110319 Triangle read by rows: T(n,k) (1 <= k <= n) is number of RNA secondary structures of size n (i.e., with n nodes) having k blocks (an RNA secondary structure can be viewed as a restricted noncrossing partition).

%I A110319 #17 Jul 06 2018 09:47:12
%S A110319 1,0,1,0,1,1,0,0,3,1,0,0,1,6,1,0,0,0,6,10,1,0,0,0,1,20,15,1,0,0,0,0,
%T A110319 10,50,21,1,0,0,0,0,1,50,105,28,1,0,0,0,0,0,15,175,196,36,1,0,0,0,0,0,
%U A110319 1,105,490,336,45,1,0,0,0,0,0,0,21,490,1176,540,55,1,0,0,0,0,0,0,1,196
%N A110319 Triangle read by rows: T(n,k) (1 <= k <= n) is number of RNA secondary structures of size n (i.e., with n nodes) having k blocks (an RNA secondary structure can be viewed as a restricted noncrossing partition).
%C A110319 Row sums yield the RNA secondary structure numbers (A004148).
%C A110319 Column sums yield the Catalan numbers (A000108).
%C A110319 A rearrangement of the Narayana numbers triangle (A001263).
%H A110319 Andrew Howroyd, <a href="/A110319/b110319.txt">Table of n, a(n) for n = 1..1275</a>
%H A110319 W. R. Schmitt and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0166-218X(92)00038-N">Linear trees and RNA secondary structure</a>, Discrete Appl. Math., 51, 317-323, 1994.
%H A110319 P. R. Stein and M. S. Waterman, <a href="http://dx.doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1978), 261-272.
%H A110319 M. Vauchassade de Chaumont and G. Viennot, <a href="http://www.emis.de/journals/SLC/opapers/s08viennot.html">Polynômes orthogonaux et problèmes d'énumeration en biologie moléculaire</a>, Publ. I.R.M.A. Strasbourg, 1984, 229/S-08, Actes 8e Sem. Lotharingien, pp. 79-86.
%F A110319 Sum_{k=1..n} k*T(n,k) = A110320(n).
%F A110319 T(n,k) = (1/k)*binomial(k, n-k)*binomial(k, n-k+1).
%F A110319 G.f.: (1 - tz - tz^2 - sqrt(1 - 2tz - 2tz^2 + t^2*z^2 - 2t^2*z^3 + t^2*z^4))/(2tz^2).
%e A110319 Triangle begins:
%e A110319   1;
%e A110319   0, 1;
%e A110319   0, 1, 1;
%e A110319   0, 0, 3, 1;
%e A110319   0, 0, 1, 6,  1;
%e A110319   0, 0, 0, 6, 10,  1;
%e A110319   0, 0, 0, 1, 20, 15,   1;
%e A110319   0, 0, 0, 0, 10, 50,  21,   1;
%e A110319   0, 0, 0, 0,  1, 50, 105,  28,  1;
%e A110319   0, 0, 0, 0,  0, 15, 175, 196, 36, 1;
%e A110319   ...
%e A110319 T(5,4)=6 because we have 13/2/4/5, 14/2/3/5. 15/2/3/4, 1/24/3/5, 1/25/3/4 and 1/2/35/4.
%p A110319 T:=(n,k)->(1/k)*binomial(k,n-k)*binomial(k,n-k+1): for n from 1 to 14 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
%t A110319 T[n_, k_] := (1/k)*Binomial[k, n - k]*Binomial[k, n - k + 1];
%t A110319 Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Jul 06 2018, from Maple *)
%o A110319 (PARI) T(n,k) = (1/k)*binomial(k, n-k)*binomial(k, n-k+1); \\ _Andrew Howroyd_, Feb 27 2018
%Y A110319 Cf. A000108, A001263, A004148, A089732, A110320.
%K A110319 nonn,tabl
%O A110319 1,9
%A A110319 _Emeric Deutsch_, Jul 19 2005