cp's OEIS Frontend

A062160 Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.

%I A062160 #42 May 31 2024 14:41:55
%S A062160 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 A062160 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 A062160 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 A062160 Square array T(n,k) = (n^k - (-1)^k)/(n+1), n >= 0, k >= 0, read by falling antidiagonals.
%C A062160 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 A062160 Seiichi Manyama, <a href="/A062160/b062160.txt">Antidiagonals n = 0..139, flattened</a>
%H A062160 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 A062160 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 A062160 T(n, k) = (-1)^(k+1) * resultant( n*x + 1, (x^k-1)/(x-1) ). - _Max Alekseyev_, Sep 28 2021
%F A062160 G.f. of row n: x/((1+x) * (1-n*x)). - _Seiichi Manyama_, Apr 12 2019
%F A062160 E.g.f. of row n: (exp(n*x) - exp(-x))/(n+1). - _Stefano Spezia_, Feb 20 2024
%F A062160 From _Peter Bala_, May 31 2024: (Start)
%F A062160 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 A062160 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 A062160 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 A062160 From _Seiichi Manyama_, Apr 12 2019: (Start)
%e A062160 Square array begins:
%e A062160    0, 1, -1,  1,  -1,    1,    -1,      1, ...
%e A062160    0, 1,  0,  1,   0,    1,     0,      1, ...
%e A062160    0, 1,  1,  3,   5,   11,    21,     43, ...
%e A062160    0, 1,  2,  7,  20,   61,   182,    547, ...
%e A062160    0, 1,  3, 13,  51,  205,   819,   3277, ...
%e A062160    0, 1,  4, 21, 104,  521,  2604,  13021, ...
%e A062160    0, 1,  5, 31, 185, 1111,  6665,  39991, ...
%e A062160    0, 1,  6, 43, 300, 2101, 14706, 102943, ... (End)
%p A062160 seq(print(seq((n^k - (-1)^k)/(n+1), k = 0..10)), n = 0..10); # _Peter Bala_, May 31 2024
%t A062160 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 A062160 Rows include A062157, A000035, A001045, A015518, A015521, A015531, A015540, A015552, A015565, A015577, A015585, A015592, A015609.
%Y A062160 Columns include A000004, A000012, A023443, A002061, A062158, A060884, A062159, A060888.
%Y A062160 Related to repunits in negative bases (cf. A055129 for positive bases).
%Y A062160 Main diagonal gives A081216.
%Y A062160 Cf. A109502.
%K A062160 sign,tabl
%O A062160 0,18
%A A062160 _Henry Bottomley_, Jun 08 2001