cp's OEIS Frontend

A142977 Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.

%I A142977 #12 Mar 15 2024 11:38:30
%S A142977 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 A142977 146,7,1,26,243,996,1765,1182,231,8,1,30,339,1804,4645,5418,2471,344,
%U A142977 9,1,34,451,2964,10165,17718,14407,4712,489,10
%N A142977 Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}.
%C A142977 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 A142977 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 A142977 T(n,k) = Sum_{j = 0..k} C(2*n, k-j)*C(2*n+j+1, j).
%F A142977 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 A142977 O.g.f. row n: (1 + x)^(2*n)/(1 - x)^(2*n+2).
%F A142977 O.g.f. column k: 1/(1 - y)*U(k, (1 + y)/(1 - y)).
%F A142977 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 A142977 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 A142977 The square array begins
%e A142977  n\k| 0...1....2.....3.....4.......5
%e A142977 ------------------------------------
%e A142977  .0.| 1...2....3.....4......5......6 ... A000027
%e A142977  .1.| 1...6...19....44.....85....146 ... A005900
%e A142977  .2.| 1..10...51...180....501...1182 ... A069038
%e A142977  .3.| 1..14...99...476...1765...5418 ... A099193
%e A142977  .4.| 1..18..163...996...4645..17718 ... A099196
%e A142977  .5.| 1..22..243..1804..10165..46530 ... A300624
%e A142977  ...
%p A142977 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 A142977 Cf. A005900 (row 1), A008288, A069038 (row 2), A099193 (row 3), A099196 (row 4), A300624 (row 5), A142978, A142983.
%K A142977 easy,nonn,tabl
%O A142977 0,3
%A A142977 _Peter Bala_, Jul 15 2008
%E A142977 Restored missing program. - _Peter Bala_, Oct 02 2008