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A270791 Triangle read by rows: coefficients of polynomials P_n(x) arising from RNA combinatorics.

%I A270791 #51 Dec 24 2018 21:25:45
%S A270791 1,1,1,158,558,135,2339,18378,13689,1575,1354,18908,28764,9660,675,
%T A270791 617926,13447818,34604118,23001156,4534875,218295,525206428,
%U A270791 16383145284,63886133214,70424606988,26926791930,3567422250,127702575,50531787,2134308548,11735772822,19350632598,12106771137,3063221550,295973325,8292375
%N A270791 Triangle read by rows: coefficients of polynomials P_n(x) arising from RNA combinatorics.
%C A270791 "... polynomials like these with nonnegative integral coefficients might reasonably be expected to be generating polynomials for some as yet unknown fatgraph structure."
%H A270791 Gheorghe Coserea, <a href="/A270791/b270791.txt">Rows n = 1..100, flattened</a>
%H A270791 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 A270791 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 A270791 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 A270791 For n = 3 we have P_3(x) = 158*x^2 + 558*x + 135.
%e A270791 For n = 4 we have P_4(x) = 2339*x^3 + 18378*x^2 + 13689*x + 1575.
%e A270791 Triangle begins:
%e A270791 n\k  [1]        [2]        [3]        [4]        [5]        [6]
%e A270791 [1]  1;
%e A270791 [2]  1,         1;
%e A270791 [3]  158        558,       135;
%e A270791 [4]  2339,      18378,     13689,     1575;
%e A270791 [5]  1354,      18908,     28764,     9660,      675;
%e A270791 [6]  617926,    13447818,  34604118,  23001156,  4534875,   218295;
%e A270791 [7]  ...
%o A270791 (PARI)
%o A270791 G = 8; N = 3*G + 1; F = 1; gmax(n) = min(n\2, G);
%o A270791 Q = matrix(N+1, G+1); Qn() = (matsize(Q)[1] - 1);
%o A270791 Qget(n, g) = { if (g < 0 || g > n/2, 0, Q[n+1, g+1]) };
%o A270791 Qset(n, g, v) = { Q[n+1, g+1] = v };
%o A270791 Quadric({x=1}) = {
%o A270791   Qset(0, 0, x);
%o A270791   for (n = 1, Qn(), for (g = 0, gmax(n),
%o A270791     my(t1 = (1+x)*(2*n-1)/3 * Qget(n-1, g),
%o A270791        t2 = (2*n-3)*(2*n-2)*(2*n-1)/12 * Qget(n-2, g-1),
%o A270791        t3 = 1/2 * sum(k = 1, n-1, sum(i = 0, g,
%o A270791        (2*k-1) * (2*(n-k)-1) * Qget(k-1, i) * Qget(n-k-1, g-i))));
%o A270791     Qset(n, g, (t1 + t2 + t3) * 6/(n+1))));
%o A270791 };
%o A270791 Quadric('x + O('x^(F+1)));
%o A270791 Kol(g) = vector(Qn()+2-F-2*g, n, polcoeff(Qget(n+F-2 + 2*g, g), F, 'x));
%o A270791 P(g) = {
%o A270791   my(x = 'x + O('x^(G+2)));
%o A270791   return(Pol(Ser(Kol(g)) * (1-4*x)^(3*g-1/2), 'x));
%o A270791 };
%o A270791 concat(vector(G, g, Vec(P(g) / content(P(g)))))  \\ _Gheorghe Coserea_, Apr 17 2016
%Y A270791 Cf. A035309, A035319, A270790.
%K A270791 nonn,easy,tabl
%O A270791 1,4
%A A270791 _N. J. A. Sloane_, Mar 28 2016
%E A270791 More terms from _Gheorghe Coserea_, Apr 17 2016