This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A109502 #24 May 31 2024 14:42:05
%S A109502 1,1,0,1,0,0,1,0,1,0,1,0,2,0,0,1,0,3,2,1,0,1,0,4,6,6,0,0,1,0,5,12,21,
%T A109502 10,1,0,1,0,6,20,52,60,22,0,0,1,0,7,30,105,204,183,42,1,0,1,0,8,42,
%U A109502 186,520,820,546,86,0,0,1,0,9,56,301,1110,2605,3276,1641,170,1,0
%N A109502 Array read by antidiagonals: T(m,n) is the number of closed walks of length n on the complete graph on m nodes, m >= 1, n >= 0.
%H A109502 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 A109502 T(m,n) = ((m-1)^n + (m-1)(-1)^n)/m.
%F A109502 G.f.: T(m, n) = [z^n](1 - (m-2)z)/(1 - (m-2)z - (m-1)z^2).
%F A109502 From _Peter Bala_, May 29 2024: (Start)
%F A109502 Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = m^(n-1) for n >= 1.
%F A109502 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, x), where o denotes the black diamond product of power series as defined by Dukes and White. Cf. A062160.
%F A109502 T(m_1*m_2, n) = Sum_{k = 0..n} Sum_{i = k..n} binomial(n, k)*binomial(n-k, i-k)*T(m_1, i)*T(m_2, n-k). (End)
%e A109502 Array begins:
%e A109502 m\n| 0 1 2 3 4 5 6 7 8 9 10
%e A109502 ---+------------------------------------------------------------
%e A109502 1 | 1 0 0 0 0 0 0 0 0 0 0
%e A109502 2 | 1 0 1 0 1 0 1 0 1 0 1
%e A109502 3 | 1 0 2 2 6 10 22 42 86 170 342
%e A109502 4 | 1 0 3 6 21 60 183 546 1641 4920 14763
%e A109502 5 | 1 0 4 12 52 204 820 3276 13108 52428 209716
%e A109502 6 | 1 0 5 20 105 520 2605 13020 65105 325520 1627605
%e A109502 7 | 1 0 6 30 186 1110 6666 39990 239946 1439670 8638026
%e A109502 8 | 1 0 7 42 301 2100 14707 102942 720601 5044200 35309407
%e A109502 9 | 1 0 8 56 456 3640 29128 233016 1864136 14913080 119304648
%e A109502 10 | 1 0 9 72 657 5904 53145 478296 4304673 38742048 348678441
%p A109502 T := proc(m, n); ((m-1)^n + (m-1)*(-1)^n)/m end:
%p A109502 seq(print(seq(T(m, n), n = 0..10)), m = 1..10); # _Peter Bala_, May 30 2024
%Y A109502 Rows are A078008, A054878, A109499, A109500, A109501.
%Y A109502 Cf. A062160.
%K A109502 nonn,easy,tabl
%O A109502 1,13
%A A109502 _Mitch Harris_, Jun 30 2005
%E A109502 Corrected by _Franklin T. Adams-Watters_, Sep 18 2006