This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A223549 #59 Sep 08 2022 08:46:04
%S A223549 1,3,1,21,15,3,77,43,35,5,1155,885,1095,315,35,4389,8589,7161,777,693,
%T A223549 63,33649,80353,42245,12285,16485,3003,231,129789,91635,233001,170145,
%U A223549 152625,20889,6435,429,4023459,3283533,9804465,8625375,9695565,1772199,819819,109395,6435,15646785,58019335,49782755,25638305,69324255,31726695,9794785,245245,230945,12155
%N A223549 Triangle T(n,k), read by rows, giving the numerator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <=n.
%C A223549 From _Petros Hadjicostas_, May 21 2020: (Start)
%C A223549 Let P_n(x) = Sum_{k=0..n} (T(n,k)/A223550(n,k))*x^k be the Boros-Moll polynomial. It follows from the theory in Comtet (1967, pp. 81-83 and 85-86) that the polynomial Q_n(x) = 2^n*n!*P_n(x) has integer coefficients and satisfies the recurrence (x-1)*Q_n(x) = 2*(2*n - 1)*(x^2 - 2)*Q_{n-1}(x) + (16*(n-1)^2 - 1)*(x + 1)*Q_{n-2}(x).
%C A223549 We have integral_{y = 0..infinity} dy/(y^4 + 2*x*y + 1)^(n + 1) = Pi * P_n(x)/(2^(n + (3/2)) * (x + 1)^(n + (1/2))) = Pi * Q_n(x)/(2^(2*n + (3/2)) * n! * (x + 1)^(n + (1/2))) for x > -1 and n integer >= 0.
%C A223549 It also follows from the theory in Comtet (1967, pp. 81-83) that g(t) = (sqrt(x + sqrt(x^2 - 1 + t)) - sqrt(x - sqrt(x^2 - 1 + t))) / sqrt((1 - t) * (x^2 - 1 + t)) = Sum_{n >= 0} t^n * P_n(x)/(2^(n - (1/2)) * (x + 1)^(n + (1/2))) for x >= 1 and 0 <= t < 1.
%C A223549 From Comtet's result, we get g(t)^2 = 2*(x - sqrt(1-t))/((1-t) * (x^2 - 1 + t)) = 2/((1-t) * (x + sqrt(1-t))) = Sum_{n >= 0} (Sum_{k=0..n} P_k(x) * P_{n-k}(x)) / (2^(n-1) * (x+1)^(n+1)) * t^n for 0 <= t < 1 and x > 1. (End)
%H A223549 Vincenzo Librandi, <a href="/A223549/b223549.txt">Rows n = 0..50, flattened</a>
%H A223549 Tewodros Amdeberhan and Victor H. Moll, <a href="http://arxiv.org/abs/0707.2118"> A formula for a quartic integral: a survey of old proofs and some new ones</a>, arXiv:0707.2118 [math.CA], 2007.
%H A223549 George Boros and Victor H. Moll, <a href="http://dx.doi.org/10.1016/S0377-0427(99)00081-3">An integral hidden in Gradshteyn and Ryzhik</a>, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
%H A223549 William Y. C. Chen and Ernest X. W. Xia, <a href="http://arxiv.org/abs/0806.4333"> The Ratio Monotonicity of the Boros-Moll Polynomials</a>, arXiv:0806.4333 [math.CO], 2009.
%H A223549 William Y. C. Chen and Ernest X. W. Xia, <a href="https://doi.org/10.1090/S0025-5718-09-02223-6"> The Ratio Monotonicity of the Boros-Moll Polynomials</a>, Mathematics of Computation, 78(268) (2009), 2269-2282.
%H A223549 Louis Comtet, <a href="https://www.jstor.org/stable/43667287">Fonctions génératrices et calcul de certaines intégrales</a>, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87.
%F A223549 T(n,k)/A223550(n,k) = 2^(-2*n)*Sum_{j=k..n} 2^j*binomial(2*n - 2*j, n - j)*binomial(n + j, j)*binomial(j, k) = 2^(-2*n)*A067001(n,n-k) for n >= 0 and k = 0..n.
%F A223549 P_n(x) = Sum_{k=0..n} (T(n, k)/A223550(n,k))*x^k = ((2*n)!/4^n/(n!)^2)*2F1([-n, n + 1], [1/2 - n], (x + 1)/2).
%F A223549 From _Petros Hadjicostas_, May 22 2020: (Start)
%F A223549 Recurrence for the polynomial: 4*n*(n - 1)*(x - 1)*P_n(x) = 4*(2*n - 1)*(n - 1)*(x^2 - 2)*P_{n-1}(x) + (16*(n - 1)^2 - 1)*(x + 1)*P_{n-2}(x).
%F A223549 O.g.f. for P_n(x): sqrt((x + 1)/(1 - 2*(x + 1)*w)/(x + sqrt(1 - 2*(x + 1)*w))). [It follows from Comtet's theory and my comments.]
%F A223549 P_n(1) = Sum_{k=0..n} T(n,k)/A223550(n,k) = A334907(n)/(2^n*n!). (End)
%e A223549 P_3(x) = 77/16 + 43*x/4 + 35*x^2/4 + 5*x^3/2.
%e A223549 As a result, integral_{y = 0..infinity} dy/(y^4 + 2*x*y + 1)^4 = Pi * P_3(x)/(2^(3 + (3/2)) * (x + 1)^(3 + (1/2))) = Pi * (40*x^3 + 140*x^2 + 172*x + 77)/(32 * sqrt(2*(x + 1)^7)) for x > -1. - _Petros Hadjicostas_, May 22 2020
%e A223549 From _Bruno Berselli_, Mar 22 2013: (Start)
%e A223549 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins as follows:
%e A223549 1;
%e A223549 3, 1;
%e A223549 21, 15, 3;
%e A223549 77, 43, 35, 5;
%e A223549 1155, 885, 1095, 315, 35;
%e A223549 4389, 8589, 7161, 777, 693, 63;
%e A223549 33649, 80353, 42245, 12285, 16485, 3003, 231;
%e A223549 129789, 91635, 233001, 170145, 152625, 20889, 6435, 429;
%e A223549 ... (End)
%t A223549 t[n_, k_] := 2^(-2*n)*Sum[ 2^j*Binomial[2*n - 2*j, n-j]*Binomial[n+j, j]*Binomial[j, k], {j, k, n}]; Table[t[n, k] // Numerator, {n, 0, 9}, {k, 0, n}] // Flatten
%o A223549 (Magma) /* As triangle: */ [[Numerator(2^(-2*n)*&+[2^j*Binomial(2*n-2*j, n-j)*Binomial(n+j, j)*Binomial(j, k): j in [k..n]]): k in [0..n]]: n in [0..10]]; // _Bruno Berselli_, Mar 22 2013
%Y A223549 Cf. A067001, A223550 (denominators), A334907.
%K A223549 nonn,easy,frac,tabl
%O A223549 0,2
%A A223549 _Jean-François Alcover_, Mar 22 2013
%E A223549 Various sections and name edited by _Petros Hadjicostas_, May 22 2020