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.

1, 6, 4, 42, 60, 24, 308, 688, 560, 160, 2310, 7080, 8760, 5040, 1120, 17556, 68712, 114576, 99456, 44352, 8064, 134596, 642824, 1351840, 1572480, 1055040, 384384, 59136, 1038312, 5864640, 14912064, 21778560, 19536000, 10695168, 3294720

Offset: 0

Author

R. J. Mathar, Mar 17 2007

Keywords

• easy
• nonn
• tabl

Comments

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).

Examples

			The table T(n,l) (with rows n >= 0 and columns l = 0..n) starts:
      1;
      6,     4;
     42,    60,     24;
    308,   688,    560,   160;
   2310,  7080,   8760,  5040,  1120;
  17556, 68712, 114576, 99456, 44352, 8064;
  ...
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
		

Links

• Tewodros Amdeberhan and Victor H. Moll, A formula for a quartic integral: a survey of old proofs and some new ones, arXiv:0707.2118 [math.CA], 2007.
• George Boros and Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, 106(2) (1999), 361-368.
• William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, arXiv:0806.4333 [math.CO], 2009.
• William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, Mathematics of Computation, 78(268) (2009), 2269-2282.
• Victor H. Moll, The evaluation of integrals: a personal story, Notices Amer. Math. Soc., 49 (No. 3, March 2002), 311-317.
• Victor H. Moll, Combinatorial sequences arising from a rational integral, Onl. J. Anal. Combin., no 2 (2007), #4.

Crossrefs

Cf. A002458 (row sums), A004982 (column l=0), A059304 (main diagonal), A067001 (rows reversed), A223549, A223550, A334907.

Programs

Maple

A126936 := proc(m, l)
    add(2^k*binomial(2*m-2*k, m-k)*binomial(m+k, m)*binomial(k, l), k=l..m):
end:
seq(seq(A126936(m,l), l=0..m), m=0..12); # R. J. Mathar, May 25 2020

Mathematica

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 *)

Formula

From Petros Hadjicostas, May 25 2020: (Start)

T(n,l) = A067001(n, n-l) = 2^(2*n) * A223549(n,l)/A223550(n,l).

Sum_{l=0..n} T(n,l) = A002458(n) = A334907(n)*2^n/n!.

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)

Extensions

Corrected by Petros Hadjicostas, May 23 2020