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 A126936 #38 May 27 2020 02:09:40
%S A126936 1,6,4,42,60,24,308,688,560,160,2310,7080,8760,5040,1120,17556,68712,
%T A126936 114576,99456,44352,8064,134596,642824,1351840,1572480,1055040,384384,
%U A126936 59136,1038312,5864640,14912064,21778560,19536000,10695168,3294720
%N A126936 Coefficients of a polynomial representation of the integral of 1/(x^4 + 2*a*x^2 + 1)^(n+1) from x = 0 to infinity.
%C A126936 The integral N(a;n) = Integral_{x=0..infinity} 1/(x^4 + 2*a*x^2 + 1)^(n+1) has a polynomial representation P_n(a) = 2^(n + 3/2) * (a+1)^(n + 1/2) * N(a;n) / Pi (known as the Boros-Moll polynomial). The table contains the coefficients T(n,l) of P_n(a) = 2^(-2*n)*Sum_{l=0..n} T(n,l)*a^l in row n and column l (with n >= 0 and 0 <= l <= n).
%H A126936 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 A126936 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 A126936 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 A126936 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 A126936 Victor H. Moll, <a href="http://www.ams.org/notices/200203/fea-moll.pdf">The evaluation of integrals: a personal story</a>, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
%H A126936 Victor H. Moll, <a href="https://web.math.rochester.edu/misc/ojac/vol2/Moll_2007.pdf">Combinatorial sequences arising from a rational integral</a>, Onl. J. Anal. Combin., no 2 (2007), #4.
%F A126936 From _Petros Hadjicostas_, May 25 2020: (Start)
%F A126936 T(n,l) = A067001(n, n-l) = 2^(2*n) * A223549(n,l)/A223550(n,l).
%F A126936 Sum_{l=0..n} T(n,l) = A002458(n) = A334907(n)*2^n/n!.
%F A126936 Bivariate o.g.f.: Sum_{n,l >= 0} T(n,l)*x^n*y^l = sqrt((1 + y)/(1 - 8*x*(1 + y))/(y + sqrt(1 - 8*x*(1 + y)))). (End)
%e A126936 The table T(n,l) (with rows n >= 0 and columns l = 0..n) starts:
%e A126936 1;
%e A126936 6, 4;
%e A126936 42, 60, 24;
%e A126936 308, 688, 560, 160;
%e A126936 2310, 7080, 8760, 5040, 1120;
%e A126936 17556, 68712, 114576, 99456, 44352, 8064;
%e A126936 ...
%e A126936 For n = 2, N(a;2) = Integral_{x=0..oo} dx/(x^4 + 2*a*x + 1)^3 = 2^(-2*2)*(Sum_{l=0..2} T(2,l)*a^l) * Pi/(2^(2 + 3/2) * (a + 1)^(2 + 1/2) = (42 + 60*a + 24*a^2) * Pi/(32 * (2*(a+1))^(5/2)) for a > -1. - _Petros Hadjicostas_, May 25 2020
%p A126936 A126936 := proc(m, l)
%p A126936 add(2^k*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m):
%p A126936 end:
%p A126936 seq(seq(A126936(m,l), l=0..m), m=0..12); # _R. J. Mathar_, May 25 2020
%t A126936 t[m_, l_] := Sum[2^k*Binomial[2*m-2*k, m-k]*Binomial[m+k, m]*Binomial[k, l], {k, l, m}]; Table[t[m, l], {m, 0, 11}, {l, 0, m}] // Flatten (* _Jean-François Alcover_, Jan 09 2014, after Maple, adapted May 2020 *)
%Y A126936 Cf. A002458 (row sums), A004982 (column l=0), A059304 (main diagonal), A067001 (rows reversed), A223549, A223550, A334907.
%K A126936 easy,nonn,tabl
%O A126936 0,2
%A A126936 _R. J. Mathar_, Mar 17 2007
%E A126936 Corrected by _Petros Hadjicostas_, May 23 2020