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%I #42 May 31 2024 14:41:55
%S 0,1,0,-1,1,0,1,0,1,0,-1,1,1,1,0,1,0,3,2,1,0,-1,1,5,7,3,1,0,1,0,11,20,
%T 13,4,1,0,-1,1,21,61,51,21,5,1,0,1,0,43,182,205,104,31,6,1,0,-1,1,85,
%U 547,819,521,185,43,7,1,0,1,0,171,1640,3277,2604,1111,300,57,8,1,0,-1,1,341,4921,13107,13021,6665,2101,455,73,9,1,0
%N Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.
%C For n >= 1, T(n, k) equals the number of walks of length k between any two distinct vertices of the complete graph K_(n+1). - _Peter Bala_, May 30 2024
%H Seiichi Manyama, <a href="/A062160/b062160.txt">Antidiagonals n = 0..139, flattened</a>
%H M. Dukes and C. D. White, <a href="http://arxiv.org/abs/1603.01589">Web Matrices: Structural Properties and Generating Combinatorial Identities</a>, arXiv:1603.01589 [math.CO], 2016.
%F T(n, k) = n^(k-1) - n^(k-2) + n^(k-3) - ... + (-1)^(k-1) = n^(k-1) - T(n, k-1) = n*T(n, k-1) - (-1)^k = (n - 1)*T(n, k-1) + n*T(n, k-2) = round[n^k/(n+1)] for n > 1.
%F T(n, k) = (-1)^(k+1) * resultant( n*x + 1, (x^k-1)/(x-1) ). - _Max Alekseyev_, Sep 28 2021
%F G.f. of row n: x/((1+x) * (1-n*x)). - _Seiichi Manyama_, Apr 12 2019
%F E.g.f. of row n: (exp(n*x) - exp(-x))/(n+1). - _Stefano Spezia_, Feb 20 2024
%F From _Peter Bala_, May 31 2024: (Start)
%F Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = (m + 1)^(n-1) for n >= 1.
%F Let R(m, x) denote the g.f. of the m-th row of the square array. Then R(m_1, x) o R(m_2, x) = R(m_1 + m_2 + m_1*m_2, x), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A109502.
%F T(m_1 + m_2 + m_1*m_2, k) = Sum_{i = 0..k} Sum_{j = i..k} binomial(k, i)* binomial(k-i, j-i)*T(m_1, j)*T(m_2, k-i). (End)
%e From _Seiichi Manyama_, Apr 12 2019: (Start)
%e Square array begins:
%e 0, 1, -1, 1, -1, 1, -1, 1, ...
%e 0, 1, 0, 1, 0, 1, 0, 1, ...
%e 0, 1, 1, 3, 5, 11, 21, 43, ...
%e 0, 1, 2, 7, 20, 61, 182, 547, ...
%e 0, 1, 3, 13, 51, 205, 819, 3277, ...
%e 0, 1, 4, 21, 104, 521, 2604, 13021, ...
%e 0, 1, 5, 31, 185, 1111, 6665, 39991, ...
%e 0, 1, 6, 43, 300, 2101, 14706, 102943, ... (End)
%p seq(print(seq((n^k - (-1)^k)/(n+1), k = 0..10)), n = 0..10); # _Peter Bala_, May 31 2024
%t T[n_,k_]:=(n^k - (-1)^k)/(n+1); Join[{0},Table[Reverse[Table[T[n-k,k],{k,0,n}]],{n,12}]]//Flatten (* _Stefano Spezia_, Feb 20 2024 *)
%Y Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592, A015609.
%Y Columns include A000004, A000012, A023443, A002061, A062158, A060884, A062159, A060888.
%Y Related to repunits in negative bases (cf. A055129 for positive bases).
%Y Main diagonal gives A081216.
%Y Cf. A109502.
%K sign,tabl
%O 0,18
%A _Henry Bottomley_, Jun 08 2001