Revision History for A173264
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| A173264
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T(0,k) = 1 and T(n,k) = [x^k] ((x - 2)*x^n + 1)/((x - 1)*(x + 1)^n) for n >= 1, square array read by descending antidiagonals (n >= 0, k >= 0).
(history;
published version)
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#10 by Peter Luschny at Wed Jan 23 08:29:37 EST 2019
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#9 by Joerg Arndt at Wed Jan 23 04:29:02 EST 2019
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#8 by Franck Maminirina Ramaharo at Wed Jan 23 04:03:22 EST 2019
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#7 by Franck Maminirina Ramaharo at Wed Jan 23 03:49:04 EST 2019
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| EXAMPLE
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------------------------------------------------
--------------------------------------------------
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#6 by Franck Maminirina Ramaharo at Wed Jan 23 03:44:10 EST 2019
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| NAME
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Antidiagonal expansion of polynomials: p(x,n)=If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/(1 + x)^n].
T(0,k) = 1 and T(n,k) = [x^k] ((x - 2)*x^n + 1)/((x - 1)*(x + 1)^n) for n >= 1, square array read by descending antidiagonals (n >= 0, k >= 0).
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| EXAMPLE
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Square array begins:
n\k | 0 1 2 3 4 5 6 7 8 ...
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0 | 1 1 1 1 1 1 1 1 1 ...
1 | -1 2 -2 2 -2 2 -2 2 -2 ...
2 | -1 1 0 -1 2 -3 4 -5 6 ...
3 | -1 2 -4 8 -14 22 -32 44 -58 ...
4 | -1 3 -7 13 -20 27 -33 37 -38 ...
5 | -1 4 -11 24 -46 82 -139 226 -354 ...
6 | -1 5 -16 40 -86 166 -294 485 -754 ...
7 | -1 6 -22 62 -148 314 -610 1108 -1910 ...
8 | -1 7 -29 91 -239 553 -1163 2269 -4164 ...
...
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| PROG
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(Maxima) (kk : 50, nn : 15)$
gf(n) := taylor(if n = 0 then 1/(1 - x) else ((x - 2)*x^n + 1)/((x - 1)*(x + 1)^n), x, 0, kk)$
T(n, k) := ratcoef(gf(n), x, k)$
create_list(T(k, n - k), n, 0, nn, k, 0, n);
/* Franck Maminirina Ramaharo, Jan 23 2019 */
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| CROSSREFS
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Cf. A173265, A173266.
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| KEYWORD
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sign,easy,tabl,uned,changed
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| EXTENSIONS
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Edited by Franck Maminirina Ramaharo, Jan 23 2019
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#5 by Franck Maminirina Ramaharo at Wed Jan 23 03:39:51 EST 2019
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| DATA
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1, 1, -1, 1, 2, -1, 1, -2, 1, -1, 1, 2, 0, 2, -1, 1, -2, -1, -4, 3, -1, 1, 2, 2, 8, -7, 4, -1, 1, -2, -3, -14, 13, -11, 5, -1, 1, 2, 4, 22, -20, 24, -16, 6, -1, 1, -2, -5, -32, 27, -46, 40, -22, 7, -1, 1, 2, 6, 44, -33, 82, -86, 62, -29, 8, -1, 1, -2, -7, -58, 37, -139, 166, -148, 91, -37, 9, -1
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| COMMENTS
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Row sums are {1, 0, 2, -1, 4, -4, 9, -12, 22, -33, ...}.
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| FORMULA
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p(x,n) = If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/(1 + x)^n];
t(n,m) = antidiagonal(p(x,n)).
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| EXAMPLE
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{1},
{1, -1},
{1, 2, -1},
{1, -2, 1, -1},
{1, 2, 0, 2, -1},
{1, -2, -1, -4, 3, -1},
{1, 2, 2, 8, -7, 4, -1},
{1, -2, -3, -14, 13, -11, 5, -1},
{ 1, 2, 4, 22, -20, 24, -16, 6, -1},
{1, -2, -5, -32, 27, -46, 40, -22, 7, -1}
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| MATHEMATICA
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Flatten[Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}]; }]]
Flatten[%]
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| KEYWORD
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sign,easy,tabl,uned
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| STATUS
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approved
editing
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#4 by Jon E. Schoenfield at Sat Dec 10 16:45:03 EST 2016
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#3 by Jon E. Schoenfield at Sat Dec 10 16:45:00 EST 2016
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| NAME
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Anti-diagonalAntidiagonal expansion of polynomials:: p(x,n)=If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/(1 + x)^n];].
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| COMMENTS
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RoeRow sums are: {1, 0, 2, -1, 4, -4, 9, -12, 22, -33, ...}.
{1, 0, 2, -1, 4, -4, 9, -12, 22, -33,...}.
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| FORMULA
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p(x,n)=) = If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/(1 + x)^n];
t(n,m)=anti-diagonal) = antidiagonal(p(x,n)))).
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| STATUS
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approved
editing
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#2 by Russ Cox at Fri Mar 30 17:34:39 EDT 2012
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| AUTHOR
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_Roger L. Bagula (rlbagulatftn(AT)yahoo.com), _, Feb 14 2010
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Discussion
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| Fri Mar 30
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| OEIS Server: https://oeis.org/edit/global/158
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#1 by N. J. A. Sloane at Tue Jun 01 03:00:00 EDT 2010
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| NAME
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Anti-diagonal expansion of polynomials:p(x,n)=If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/(1 + x)^n];
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| DATA
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1, 1, -1, 1, 2, -1, 1, -2, 1, -1, 1, 2, 0, 2, -1, 1, -2, -1, -4, 3, -1, 1, 2, 2, 8, -7, 4, -1, 1, -2, -3, -14, 13, -11, 5, -1, 1, 2, 4, 22, -20, 24, -16, 6, -1, 1, -2, -5, -32, 27, -46, 40, -22, 7, -1
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| OFFSET
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0,5
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| COMMENTS
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Roe sums are:
{1, 0, 2, -1, 4, -4, 9, -12, 22, -33,...}.
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| FORMULA
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p(x,n)=If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/(1 + x)^n];
t(n,m)=anti-diagonal(p(x,n))
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| EXAMPLE
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{1},
{1, -1},
{1, 2, -1},
{1, -2, 1, -1},
{1, 2, 0, 2, -1},
{1, -2, -1, -4, 3, -1},
{1, 2, 2, 8, -7, 4, -1},
{1, -2, -3, -14, 13, -11, 5, -1},
{ 1, 2, 4, 22, -20, 24, -16, 6, -1},
{1, -2, -5, -32, 27, -46, 40, -22, 7, -1}
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| MATHEMATICA
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p[x_, n_] = If[n == 0, 1/(1 - x), (x^n - Sum[x^i, {i, 0, n - 1}])/( 1 + x)^n];
a = Table[Table[SeriesCoefficient[Series[p[x, n], {x, 0, 50}], m], {m, 0, 20}], {n, 0, 20}];
Table[Table[a[[m, n - m + 1]], {m, 1, n}], {n, 1, 10}];
Flatten[%]
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| KEYWORD
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sign,tabl,uned
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| AUTHOR
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Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 14 2010
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| STATUS
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approved
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