|
|
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.
|
|
6
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
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
|
|
LINKS
|
|
|
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).
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)
|
|
EXAMPLE
|
P_3(x) = 77/16 + 43*x/4 + 35*x^2/4 + 5*x^3/2.
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)
|
|
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
|
|
PROG
|
(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
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|