A270791 Triangle read by rows: coefficients of polynomials P_n(x) arising from RNA combinatorics.
1, 1, 1, 158, 558, 135, 2339, 18378, 13689, 1575, 1354, 18908, 28764, 9660, 675, 617926, 13447818, 34604118, 23001156, 4534875, 218295, 525206428, 16383145284, 63886133214, 70424606988, 26926791930, 3567422250, 127702575, 50531787, 2134308548, 11735772822, 19350632598, 12106771137, 3063221550, 295973325, 8292375
Offset: 1
Examples
For n = 3 we have P_3(x) = 158*x^2 + 558*x + 135. For n = 4 we have P_4(x) = 2339*x^3 + 18378*x^2 + 13689*x + 1575. Triangle begins: n\k [1] [2] [3] [4] [5] [6] [1] 1; [2] 1, 1; [3] 158 558, 135; [4] 2339, 18378, 13689, 1575; [5] 1354, 18908, 28764, 9660, 675; [6] 617926, 13447818, 34604118, 23001156, 4534875, 218295; [7] ...
Links
- Gheorghe Coserea, Rows n = 1..100, flattened
- J. E. Andersen, R. C. Penner, C. M. Reidys, M. S. Waterman, Topological classification and enumeration of RNA structures by genus, J. Math. Biol. 65 (2013) 1261-1278
- R. C. Penner, Moduli Spaces and Macromolecules, Bull. Amer. Math. Soc., 53 (2015), 217-268. See p. 259.
Programs
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PARI
G = 8; N = 3*G + 1; F = 1; gmax(n) = min(n\2, G); Q = matrix(N+1, G+1); Qn() = (matsize(Q)[1] - 1); Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) }; Qset(n, g, v) = { Q[n+1, g+1] = v }; Quadric({x=1}) = { Qset(0, 0, x); for (n = 1, Qn(), for (g = 0, gmax(n), my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g), t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1), t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g, (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i)))); Qset(n, g, (t1 + t2 + t3) * 6/(n+1)))); }; Quadric('x + O('x^(F+1))); Kol(g) = vector(Qn()+2-F-2*g, n, polcoeff(Qget(n+F-2 + 2*g, g), F, 'x)); P(g) = { my(x = 'x + O('x^(G+2))); return(Pol(Ser(Kol(g)) * (1-4*x)^(3*g-1/2), 'x)); }; concat(vector(G, g, Vec(P(g) / content(P(g))))) \\ Gheorghe Coserea, Apr 17 2016
Formula
The g.f. for column g>0 of triangle A035309 is x^(2*g) * A270790(g) * P_g(x) / (1-4*x)^(3*g-1/2), where P_g(x) is the polynomial associated with row g of the triangle. - Gheorghe Coserea, Apr 17 2016
Extensions
More terms from Gheorghe Coserea, Apr 17 2016
Comments