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Triangle read by rows: coefficients of polynomials P_n(x) arising from RNA combinatorics.
3

%I #51 Dec 24 2018 21:25:45

%S 1,1,1,158,558,135,2339,18378,13689,1575,1354,18908,28764,9660,675,

%T 617926,13447818,34604118,23001156,4534875,218295,525206428,

%U 16383145284,63886133214,70424606988,26926791930,3567422250,127702575,50531787,2134308548,11735772822,19350632598,12106771137,3063221550,295973325,8292375

%N Triangle read by rows: coefficients of polynomials P_n(x) arising from RNA combinatorics.

%C "... polynomials like these with nonnegative integral coefficients might reasonably be expected to be generating polynomials for some as yet unknown fatgraph structure."

%H Gheorghe Coserea, <a href="/A270791/b270791.txt">Rows n = 1..100, flattened</a>

%H J. E. Andersen, R. C. Penner, C. M. Reidys, M. S. Waterman, <a href="https://doi.org/10.1007/s00285-012-0594-x">Topological classification and enumeration of RNA structures by genus</a>, J. Math. Biol. 65 (2013) 1261-1278

%H R. C. Penner, <a href="http://dx.doi.org/10.1090/bull/1524">Moduli Spaces and Macromolecules</a>, Bull. Amer. Math. Soc., 53 (2015), 217-268. See p. 259.

%F 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

%e For n = 3 we have P_3(x) = 158*x^2 + 558*x + 135.

%e For n = 4 we have P_4(x) = 2339*x^3 + 18378*x^2 + 13689*x + 1575.

%e Triangle begins:

%e n\k [1] [2] [3] [4] [5] [6]

%e [1] 1;

%e [2] 1, 1;

%e [3] 158 558, 135;

%e [4] 2339, 18378, 13689, 1575;

%e [5] 1354, 18908, 28764, 9660, 675;

%e [6] 617926, 13447818, 34604118, 23001156, 4534875, 218295;

%e [7] ...

%o (PARI)

%o G = 8; N = 3*G + 1; F = 1; gmax(n) = min(n\2, G);

%o Q = matrix(N+1, G+1); Qn() = (matsize(Q)[1] - 1);

%o Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };

%o Qset(n, g, v) = { Q[n+1, g+1] = v };

%o Quadric({x=1}) = {

%o Qset(0, 0, x);

%o for (n = 1, Qn(), for (g = 0, gmax(n),

%o my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),

%o t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),

%o t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,

%o (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));

%o Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));

%o };

%o Quadric('x + O('x^(F+1)));

%o Kol(g) = vector(Qn()+2-F-2*g, n, polcoeff(Qget(n+F-2 + 2*g, g), F, 'x));

%o P(g) = {

%o my(x = 'x + O('x^(G+2)));

%o return(Pol(Ser(Kol(g)) * (1-4*x)^(3*g-1/2), 'x));

%o };

%o concat(vector(G, g, Vec(P(g) / content(P(g))))) \\ _Gheorghe Coserea_, Apr 17 2016

%Y Cf. A035309, A035319, A270790.

%K nonn,easy,tabl

%O 1,4

%A _N. J. A. Sloane_, Mar 28 2016

%E More terms from _Gheorghe Coserea_, Apr 17 2016