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Search: id:A142977
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| A142977 |
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Table of coefficients in the expansion of the rational function 1/{(1-x)^2 - y*(1+x)^2}. |
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+0
3
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| 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, 146, 7, 1, 26, 243, 996, 1765, 1182, 231, 8, 1, 30, 339, 1804, 4645, 5418, 2471, 344, 9, 1, 34, 451, 2964, 10165, 17718, 14407, 4712, 489, 10
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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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.
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LINKS
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Hyun Kwang Kim, On regular polytope numbers
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FORMULA
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T(n,k) = sum {j = 0..k} C(2*n,k-j)*C(2*n+j+1,j). 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..inf} U(m,(1+y)/(1-y))*x^m, where U(m,y) denotes the m_th Chebyshev polynomial of the second kind. O.g.f. row n: (1+x)^(2*n)/(1-x)^(2*n+2). O.g.f. column k: 1/(1-y)*U(k,(1+y)/(1-y)). The entries in the n_th row appear in the series acceleration formula for the constant log(2): sum {k = 1..inf} (-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)]. 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.
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EXAMPLE
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The square array begins
n\k|0...1....2.....3.....4.......5
---------------------------------------------
.0.|1...2....3.....4......5......6... A000027
.1.|1...6...19....44.....85....146... A005900
.2.|1..10...51...180....501...1182... A069038
.3.|1..14...99...476...1765...5418... A099193
.4.|1..18..163...996...4645..17718... A099196
.5.|1..22..243..1804..10165..46530
...
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MAPLE
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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;
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CROSSREFS
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Cf. A005900 (row 1), A008288, A069038 (row 2), A099193 (row 3), A099196 (row 4), A142978, A142983.
Sequence in context: A006019 A065553 A016545 this_sequence A120108 A060556 A132813
Adjacent sequences: A142974 A142975 A142976 this_sequence A142978 A142979 A142980
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Peter Bala (pbala(AT)toucansurf.com), Jul 15 2008
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EXTENSIONS
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Restored missing program. - Peter Bala (pbala(AT)toucansurf.com), Oct 02 2008
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