A109502 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

1,13

LINKS

Table of n, a(n) for n=1..78.

M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.

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)

EXAMPLE

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

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

CROSSREFS

Rows are A078008, A054878, A109499, A109500, A109501.

Cf. A062160.

Sequence in context: A306605 A363929 A054876 * A323887 A349913 A346236

Adjacent sequences: A109499 A109500 A109501 * A109503 A109504 A109505

KEYWORD

nonn,easy,tabl

AUTHOR

Mitch Harris, Jun 30 2005

EXTENSIONS

Corrected by Franklin T. Adams-Watters, Sep 18 2006

STATUS

approved

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