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A202841 Triangle read by rows: T(n,k) is the number of secondary structures of size n having k stacks of length 2 (n>=0, k>=0). 6
1, 1, 1, 2, 4, 7, 1, 14, 3, 31, 6, 66, 16, 142, 43, 316, 104, 3, 708, 256, 14, 1593, 647, 43, 3625, 1610, 138, 8314, 3990, 430, 1, 19165, 9944, 1247, 16, 44433, 24762, 3552, 85, 103557, 61574, 10040, 331, 242376, 153270, 27877, 1225, 569514, 381718, 76491, 4272, 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).

LINKS

Table of n, a(n) for n=0..51.

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

Sum(k*T(n,k), k>=0) = A202839(n-2).

T(n,0) = A202842(n).

G.f. G(t,z) satisfies G = 1 + zG + [f/(1 + f)]G(G-1), where f = (t-1)z^4 + z^2/(1-z^2).

The multivariate g.f. H(z, t[1], t[2], ...) 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 = t[1]z^2 + t[2]z^4 + t[3]z^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; only the last one has a stack of length 2.

Triangle starts:

1;

1;

1;

2;

4;

7,1;

14,3;

31,6;

MAPLE

f := (t-1)*z^4+z^2/(1-z^2): eq := G = 1+z*G+f*G*(G-1)/(1+f): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 23)): 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. A004148, A202838, A202839, A202840, A202842, A202843, A202844

Sequence in context: A118390 A247294 A202848 * A247290 A246183 A134974

Adjacent sequences:  A202838 A202839 A202840 * A202842 A202843 A202844

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Dec 25 2011

STATUS

approved

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Last modified June 6 18:59 EDT 2020. Contains 334832 sequences. (Running on oeis4.)