login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A223549 Triangle T(n,k) giving numerator of coefficient of x^k in Boros-Moll polynomial P_n(x^k) n >= 0, 0<=k<=n. 2
1, 3, 1, 21, 15, 3, 77, 43, 35, 5, 1155, 885, 1095, 315, 35, 4389, 8589, 7161, 777, 693, 63, 33649, 80353, 42245, 12285, 16485, 3003, 231, 129789, 91635, 233001, 170145, 152625, 20889, 6435, 429, 4023459, 3283533, 9804465, 8625375, 9695565, 1772199, 819819, 109395, 6435, 15646785, 58019335, 49782755, 25638305, 69324255, 31726695, 9794785, 245245, 230945, 12155 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Coefficients can also be evaluated as: t(n, k) = (2*n)!/4^n/n!^2*Coefficient( Series( Hypergeometric2F1(-n, n+1, 1/2-n, (x+1)/2), {x, 0, n}), x, k).

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, 2007, arXiv:0707.2118 [math.CA]

George Boros, Victor H. Moll, An integral hidden in Gradshteyn and Ryzhik, Journal of Computational and Applied Mathematics, Volume 106, Issue 2, 30 June 1999, Pages 361-368.

William Y. C. Chen and Ernest X. W. Xia, The Ratio Monotonicity of the Boros-Moll Polynomials, 2009, arXiv:0806.4333 [math.CO]

EXAMPLE

P_3(x) = 77/16 + 43x/4 + 35x^2/4 + 5x^3/2.

From Bruno Berselli, Mar 22 2013: (Start)

Triangle begins:

1;

3, 1;

21, 15, 3;

77, 43, 35, 5;

1155, 885, 1095, 315, 35;

4389, 8589, 7161, 777, 693, 63;

33649, 80353, 42245, 12285, 16485, 3003, 231;

129789, 91635, 233001, 170145, 152625, 20889, 6435, 429, etc. (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] // Numerator, {n, 0, 9}, {k, 0, n}] // Flatten

PROG

(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

CROSSREFS

Cf. A223550 (denominators).

Sequence in context: A225471 A136236 A113090 * A138354 A193632 A190962

Adjacent sequences:  A223546 A223547 A223548 * A223550 A223551 A223552

KEYWORD

nonn,easy,frac,tabl

AUTHOR

Jean-Fran├žois Alcover, Mar 22 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 20 00:47 EDT 2019. Contains 324223 sequences. (Running on oeis4.)