A212801 Square array read by antidiagonals: T(m,n) = number of Eulerian circuits in the Cartesian product of two directed cycles of lengths m and n.
1, 1, 1, 1, 4, 1, 1, 13, 13, 1, 1, 40, 108, 40, 1, 1, 121, 793, 793, 121, 1, 1, 364, 5611, 12800, 5611, 364, 1, 1, 1093, 39312, 193721, 193721, 39312, 1093, 1, 1, 3280, 274933, 2886520, 6050000, 2886520, 274933, 3280, 1, 1, 9841, 1923025, 42999713, 183990301, 183990301, 42999713, 1923025, 9841, 1
Offset: 1
Examples
Array begins: ====================================================================== m\n| 1 2 3 4 5 6 7 ---|------------------------------------------------------------------ 1 | 1 1 1 1 1 1 1 ... 2 | 1 4 13 40 121 364 1093 ... 3 | 1 13 108 793 5611 39312 274933 ... 4 | 1 40 793 12800 193721 2886520 42999713 ... 5 | 1 121 5611 193721 6050000 183990301 5598183221 ... 6 | 1 364 39312 2886520 183990301 11218701312 681838513861 ... 7 | 1 1093 274933 42999713 5598183221 681838513861 81959473720768 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275
- Germain Kreweras, Complexité et circuits Eulériens dans les sommes tensorielles de graphes, J. Combin. Theory, B 24 (1978), 202-212. See p. 211.
Crossrefs
Programs
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Mathematica
T[m_, n_] := Product[2 - Exp[2*I*h*Pi/m] - Exp[2*I*k*Pi/n], {h, 1, m - 1}, {k, 1, n - 1}]; Table[T[m - n + 1, n] // FullSimplify, {m, 1, 10}, {n, 1, m}] // Flatten (* Jean-François Alcover, Jun 30 2018 *) -
PARI
T(m,n) = prod(k=1, n-1, prod(h=1, m-1, 2 - exp(2*I*h*Pi/m) - exp(2*I*k*Pi/n))); tabl(nn) = matrix(nn, nn, m, n, round(real(T(m,n)))); \\ Michel Marcus, Feb 01 2016
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PARI
\\ all integer version. R(n,f)={my(p=lift(prod(i=1, n-1, f(Mod(x^i, 1-x^n))))); sumdiv(n, d, moebius(n/d) * polcoef(p, d%n, x))} T(m,n)={my(p=R(n, x->2-x-y)); R(m, x->subst(p, y, x))} \\ Andrew Howroyd, May 19 2020
Formula
T(m,n) = Product_{k=1..n-1} Product_{h=1..m-1} (2 - exp(2*i*h*Pi/m) - exp(2*i*k*Pi/n)), where i is the imaginary unit.
Extensions
Name clarified by Andrew Howroyd, Jan 12 2018
Comments