A223550 Triangle T(n,k), read by rows, giving the denominator of the coefficient of x^k in the Boros-Moll polynomial P_n(x) for n >= 0 and 0 <= k <= n.

1, 2, 1, 8, 4, 2, 16, 4, 4, 2, 128, 32, 32, 16, 8, 256, 128, 64, 8, 16, 8, 1024, 512, 128, 32, 64, 32, 16, 2048, 256, 256, 128, 128, 32, 32, 16, 32768, 4096, 4096, 2048, 2048, 512, 512, 256, 128, 65536, 32768, 8192, 2048, 4096, 2048, 1024, 64, 256, 128

Offset: 0

Author

Jean-François Alcover, Mar 22 2013

Keywords

• nonn
• easy
• frac
• tabl

Comments

As Chen and Xia (2009) state, the Boros-Moll polynomial P_n(x) can be viewed as a Jacobi polynomial P_n^{a,b}(x) with a = n + (1/2) and b = -(n + (1/2)). For more information about the relation of this polynomial P_n(x) to the theory in Comtet (1967, pp. 81-83 and 85-86), see my comments for A223549. - Petros Hadjicostas, May 22 2020

Examples

			P_3(x) = 77/16 + 43*x/4 + 35*x^2/4 + 5*x^3/2.
From _Bruno Berselli_, Mar 22 2013: (Start)
Triangle T(n,k) (with rows n >= 0 and columns k=0..n) begins as follows:
      1;
      2,     1;
      8,     4,    2;
     16,     4,    4,    2;
    128,    32,   32,   16,    8;
    256,   128,   64,    8,   16,    8;
   1024,   512,  128,   32,   64,   32,   16;
   2048,   256,  256,  128,  128,   32,   32,  16;
  32768,  4096, 4096, 2048, 2048,  512,  512, 256, 128;
  65536, 32768, 8192, 2048, 4096, 2048, 1024,  64, 256, 128;
  ... (End)
		

Links

• Vincenzo Librandi, Rows n = 0..50, flattened
• 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.
• Louis Comtet, Fonctions génératrices et calcul de certaines intégrales, Publikacije Elektrotechnickog faculteta - Serija Matematika i Fizika, No. 181/196 (1967), 77-87.

Crossrefs

Cf. A067001, A223549 (numerators), A334907.

Programs

Magma

/* As triangle: */ [[Denominator(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

Mathematica

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] // Denominator, {n, 0, 9}, {k, 0, n}] // Flatten

Formula

A223549(n,k)/T(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.

P_n(x) = Sum_{k=0..n} (A223549(n,k)/T(n,k))*x^k = ((2*n)!/4^n/(n!)^2)*2F1([-n, n + 1], [1/2 - n], (x + 1)/2).

From Petros Hadjicostas, May 22 2020: (Start)

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).

P_n(1) = Sum_{k=0..n} A223549(n,k)/T(n,k) = A334907(n)/(2^n*n!). (End)

Extensions

Name edited by Petros Hadjicostas, May 22 2020