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 A270791 Triangle read by rows: coefficients of polynomials P_n(x) arising from RNA combinatorics. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS "... polynomials like these with nonnegative integral coefficients might reasonably be expected to be generating polynomials for some as yet unknown fatgraph structure." 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. 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 EXAMPLE 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] ... PROG (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 CROSSREFS Cf. A035309, A035319, A270790. Sequence in context: A072555 A056088 A189813 * A180096 A250996 A280483 Adjacent sequences: A270788 A270789 A270790 * A270792 A270793 A270794 KEYWORD nonn,easy,tabl AUTHOR N. J. A. Sloane, Mar 28 2016 EXTENSIONS More terms from Gheorghe Coserea, Apr 17 2016 STATUS approved

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Last modified November 30 02:55 EST 2023. Contains 367452 sequences. (Running on oeis4.)