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.
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, 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, 186, 520, 820, 546, 86, 0, 0, 1, 0, 9, 56, 301, 1110, 2605, 3276, 1641, 170, 1, 0
Offset: 1
Examples
Array begins: m\n| 0 1 2 3 4 5 6 7 8 9 10 ---+------------------------------------------------------------ 1 | 1 0 0 0 0 0 0 0 0 0 0 2 | 1 0 1 0 1 0 1 0 1 0 1 3 | 1 0 2 2 6 10 22 42 86 170 342 4 | 1 0 3 6 21 60 183 546 1641 4920 14763 5 | 1 0 4 12 52 204 820 3276 13108 52428 209716 6 | 1 0 5 20 105 520 2605 13020 65105 325520 1627605 7 | 1 0 6 30 186 1110 6666 39990 239946 1439670 8638026 8 | 1 0 7 42 301 2100 14707 102942 720601 5044200 35309407 9 | 1 0 8 56 456 3640 29128 233016 1864136 14913080 119304648 10 | 1 0 9 72 657 5904 53145 478296 4304673 38742048 348678441
Links
- M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Programs
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Maple
T := proc(m, n); ((m-1)^n + (m-1)*(-1)^n)/m end: seq(print(seq(T(m, n), n = 0..10)), m = 1..10); # Peter Bala, May 30 2024
Formula
T(m,n) = ((m-1)^n + (m-1)(-1)^n)/m.
G.f.: T(m, n) = [z^n](1 - (m-2)z)/(1 - (m-2)z - (m-1)z^2).
From Peter Bala, May 29 2024: (Start)
Binomial transform of the m-th row: Sum_{k = 0..n} binomial(n, k)*T(m, k) = m^(n-1) for n >= 1.
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.
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)
Extensions
Corrected by Franklin T. Adams-Watters, Sep 18 2006