The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A202848 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of even length (n>=0, k>=0). 3
 1, 1, 1, 2, 4, 7, 1, 14, 3, 31, 6, 66, 16, 141, 44, 313, 107, 3, 702, 262, 14, 1577, 663, 43, 3581, 1654, 138, 8207, 4091, 436, 1, 18903, 10178, 1275, 16, 43770, 25339, 3638, 85, 101903, 62952, 10316, 331, 238282, 156495, 28743, 1228, 559322, 389374, 78979, 4320, 9 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209. Sum of entries in row n is A004148 (the secondary structure numbers). Sum(k*T(n,k), k>=0) = A202846(n-2). T(n,0) = A202849(n). LINKS I. L. Hofacker, P. Schuster and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237. P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272. FORMULA G.f.: G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (z^2 + t*z^4)/(1-z^4). The multivariate g.f. H(z, t, t, ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + (f/(1 + f))H(H-1), where f = tz^2 + tz^4 + tz^6 + ... . EXAMPLE Row 5 is 7,1: representing unpaired vertices by v and arcs by AA, BB, etc., the 8 (= A004148(5)) secondary structures of size 5 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; ABVBA; the last one has 1 stack of length 2. Triangle starts: 1; 1; 1; 2; 4; 7,1; 14,3; 31,6; MAPLE f := (z^2+t*z^4)/(1-z^4): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 25)): for n from 0 to 19 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 19 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form CROSSREFS Cf. A004145, A202846, A023427, A202849, Sequence in context: A098073 A118390 A247294 * A202841 A247290 A246183 Adjacent sequences:  A202845 A202846 A202847 * A202849 A202850 A202851 KEYWORD nonn,tabf AUTHOR Emeric Deutsch, Dec 26 2011 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified July 12 11:25 EDT 2020. Contains 335658 sequences. (Running on oeis4.)