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Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.
2

%I #12 Mar 15 2024 11:38:30

%S 1,1,2,1,6,3,1,10,19,4,1,14,51,44,5,1,18,99,180,85,6,1,22,163,476,501,

%T 146,7,1,26,243,996,1765,1182,231,8,1,30,339,1804,4645,5418,2471,344,

%U 9,1,34,451,2964,10165,17718,14407,4712,489,10

%N Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.

%C The row entries are the figurate numbers of the odd dimensional cross polytopes. See A142978 for the complete table of figurate numbers of n-dimensional cross polytopes. The rows are the partial sums of the even-numbered rows of the square array of Delannoy numbers A008288.

%H Hyun Kwang Kim, <a href="http://dx.doi.org/10.1090/S0002-9939-02-06710-2">On Regular Polytope Numbers</a>, Proc. Amer. Math. Soc., 131 (2003), 65-75.

%F T(n,k) = Sum_{j = 0..k} C(2*n, k-j)*C(2*n+j+1, j).

%F O.g.f.: 1/{(1 - x)^2 - y*(1 + x)^2} = Sum_{n, k >= 0} T(n,k)*x^k*y^n = 1/(1 - y) * Sum_{m >= 0} U(m, (1 + y)/(1 - y))*x^m, where U(m, y) denotes the m-th Chebyshev polynomial of the second kind.

%F O.g.f. row n: (1 + x)^(2*n)/(1 - x)^(2*n+2).

%F O.g.f. column k: 1/(1 - y)*U(k, (1 + y)/(1 - y)).

%F The entries in the n-th row appear in the series acceleration formula for the constant log(2): Sum_{k >= 1} (-1)^(k+1)/(T(n,k)*T(n,k+1)) = 1 + (4*n + 2)*( log(2) - (1 - 1/2 + 1/3 - ... + 1/(2*n + 1)) ).

%F For example, n = 1 gives log(2) = 4/6 + (1/6)*( 1/(1*6) - 1/(6*19) + 1/(19*44) - 1/(44*85) + ... ). See A142983 for further details.

%e The square array begins

%e n\k| 0...1....2.....3.....4.......5

%e ------------------------------------

%e .0.| 1...2....3.....4......5......6 ... A000027

%e .1.| 1...6...19....44.....85....146 ... A005900

%e .2.| 1..10...51...180....501...1182 ... A069038

%e .3.| 1..14...99...476...1765...5418 ... A099193

%e .4.| 1..18..163...996...4645..17718 ... A099196

%e .5.| 1..22..243..1804..10165..46530 ... A300624

%e ...

%p with(combinat): T:=(n,k) -> add(binomial(2n,k-j)*binomial(2n+j+1,j), j = 0..k): for n from 0 to 9 do seq(T(n,k), k = 0..9) end do;

%Y Cf. A005900 (row 1), A008288, A069038 (row 2), A099193 (row 3), A099196 (row 4), A300624 (row 5), A142978, A142983.

%K easy,nonn,tabl

%O 0,3

%A _Peter Bala_, Jul 15 2008

%E Restored missing program. - _Peter Bala_, Oct 02 2008