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%I #72 Feb 16 2021 01:51:14
%S 1,0,1,0,1,1,0,2,2,1,0,4,5,3,1,0,8,12,9,4,1,0,16,28,25,14,5,1,0,32,64,
%T 66,44,20,6,1,0,64,144,168,129,70,27,7,1,0,128,320,416,360,225,104,35,
%U 8,1,0,256,704,1008,968,681,363,147,44,9,1,0,512,1536,2400,2528,1970
%N A(n,k) is the coefficient of x^k in (1-x)^n/(1-2*x)^n for n, k >= 0; Table A read by descending antidiagonals.
%C The triangular version of this square array is defined by T(n,k) = A(k,n-k) for 0 <= k <= n. Conversely, A(n,k) = T(n+k,n) for n,k >= 0. We have [o.g.f of T](x,y) = [o.g.f. of A](x*y, x) and [o.g.f. of A](x,y) = [o.g.f. of T](y,x/y). - Petros Hadjicostas, Feb 11 2021
%C From _Paul Barry_, Nov 10 2008: (Start)
%C As number triangle, Riordan array (1, x(1-x)/(1-2x)). A062110*A007318 is A147703.
%C [0,1,1,0,0,0,....] DELTA [1,0,0,0,.....]. (_Philippe Deléham_'s DELTA is defined in A084938.) (End)
%C Modulo 2, this triangle T becomes triangle A106344. - _Philippe Deléham_, Dec 18 2008
%H Michael De Vlieger, <a href="/A062110/b062110.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened.)
%H Eunice Y. S. Chan, Robert M. Corless, Laureano Gonzalez-Vega, J. Rafael Sendra, and Juana Sendra, <a href="https://arxiv.org/abs/1907.10677">Upper Hessenberg and Toeplitz Bohemians</a>, arXiv:1907.10677 [cs.SC], 2019.
%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq., 21 (2018), #18.1.4.
%F Formulas for the square array (A(n,k): n,k >= 0):
%F A(n, k) = A(n-1, k) + Sum_{0 <= j < k} A(n, j) for n >= 1 and k >= 0 with A(0, k) = 0^k for k >= 0.
%F G.f.: 1/(1-x*(1-y)/(1-2*y)) = Sum_{i, j >= 0} A(i, j) x^i*y^j.
%F From _Petros Hadjicostas_, Feb 15 2021: (Start)
%F A(n,k) = 2^(k-n)*n*hypergeom([1-n, k+1], [2], -1) for n >= 0 and k >= 1.
%F A(n,k) = 2*A(n,k-1) + A(n-1,k) - A(n-1,k-1) for n,k >= 1 with A(n,0) = 1 for n >= 0 and A(0,k) = 0 for k >= 1. (End)
%F Formulas for the triangle (T(n,k): 0 <= k <= n):
%F From _Philippe Deléham_, Aug 01 2006: (Start)
%F T(n,k) = A121462(n+1,k+1)*2^(n-2*k) for 0 <= k < n.
%F T(n,k) = 2^(n-2*k)*k*hypergeom([1-k, n-k+1], [2], -1) for 0 <= k < n. (End)
%F Sum_{k=0..n} T(n,k)*x^k = A152239(n), A152223(n), A152185(n), A152174(n), A152167(n), A152166(n), A152163(n), A000007(n), A001519(n), A006012(n), A081704(n), A082761(n), A147837(n), A147838(n), A147839(n), A147840(n), A147841(n), for x = -7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. - _Philippe Deléham_, Dec 09 2008
%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) for 1 <= k <= n-1 with T(0,0) = T(1,1) = T(2,1) = T(2,2) = 1, T(1,0) = T(2,0) = 0, and T(n,k) = 0 if k > n or if k < 0. - _Philippe Deléham_, Oct 30 2013
%F G.f.: Sum_{n.k>=0} T(n,k)*x^n*y^k = (1 - 2*x)/(x^2*y - x*y - 2*x + 1). - _Petros Hadjicostas_, Feb 15 2021
%e Table A(n,k) (with rows n >= 0 and columns k >= 0) begins:
%e 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
%e 1, 1, 2, 4, 8, 16, 32, 64, 128, 256, ...
%e 1, 2, 5, 12, 28, 64, 144, 320, 704, 1536, ...
%e 1, 3, 9, 25, 66, 168, 416, 1008, 2400, 5632, ...
%e 1, 4, 14, 44, 129, 360, 968, 2528, 6448, 16128, ...
%e 1, 5, 20, 70, 225, 681, 1970, 5500, 14920, 39520, ...
%e 1, 6, 27, 104, 363, 1182, 3653, 10836, 31092, 86784, ...
%e ... - _Petros Hadjicostas_, Feb 15 2021
%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 2, 1;
%e 0, 4, 5, 3, 1;
%e 0, 8, 12, 9, 4, 1;
%e 0, 16, 28, 25, 14, 5, 1;
%e 0, 32, 64, 66, 44, 20, 6, 1;
%e 0, 64, 144, 168, 129, 70, 27, 7, 1;
%e 0, 128, 320, 416, 360, 225, 104, 35, 8, 1;
%e ... - _Philippe Deléham_, Nov 30 2008
%t t[n_, n_] = 1; t[n_, k_] := 2^(n-2*k)*k*Hypergeometric2F1[1-k, n-k+1, 2, -1]; Table[t[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 30 2013, after _Philippe Deléham_ + symbolic sum *)
%o (PARI) a(i,j)=if(i<0 || j<0,0,polcoeff(((1-x)/(1-2*x)+x*O(x^j))^i,j))
%Y Rows of A include A000007, A011782, A045623, A058396, A062109, A169792, A169793, A169794, A169795, A169796, A169797.
%Y Columns of A include A000012, A001477, A000096, A000297.
%Y Main diagonal of A is A002002.
%Y Table A(n, k) is a multiple of 2^(k-n); dividing by this gives a table similar to A050143 except at the edges.
%Y Essentially the same array as A105306, A160232.
%K nonn,tabl
%O 0,8
%A _Henry Bottomley_, May 30 2001
%E Various sections edited by _Petros Hadjicostas_, Feb 15 2021
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