A307468
Cogrowth sequence for the Heisenberg group.
Original entry on oeis.org
1, 4, 28, 232, 2156, 21944, 240280, 2787320, 33820044, 424925872, 5486681368, 72398776344, 972270849512, 13247921422480, 182729003683352, 2546778437385032, 35816909974343308, 507700854900783784, 7246857513425470288, 104083322583897779656
Offset: 0
For n=1 the a(1)=4 words are x^{-1}x, xx^{-1}, y^{-1}y, yy^{-1}.
- Jay Pantone, Table of n, a(n) for n = 0..200
- Cédric Béguin, Alain Valette and Andrzej Zuk, On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, Journal of Geometry and Physics, 21 (1997), 337-356.
- D. Lind and K, Schmidt, A survey of algebraic actions of the discrete Heisenberg group, arXiv:1502.06243 [math.DS], 2015; Russian Mathematical Surveys, 70:4 (2015), 77-142.
Related cogrowth sequences: Z
A000984, Z^2
A002894, Z^3
A002896, (Z/kZ)^*2 for k = 2..5:
A126869,
A047098,
A107026,
A304979, Richard Thompson's group F
A246877. The cogrowth sequences for BS(N,N) for N = 2..10 are
A229644,
A229645,
A229646,
A229647,
A229648,
A229649,
A229650,
A229651,
A229652.
A382100
Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of 1/(2 - B_k(x)), where B_k(x) = 1 + x*B_k(x)^k.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 4, 1, 1, 1, 4, 10, 8, 1, 1, 1, 5, 19, 35, 16, 1, 1, 1, 6, 31, 98, 126, 32, 1, 1, 1, 7, 46, 213, 531, 462, 64, 1, 1, 1, 8, 64, 396, 1556, 2974, 1716, 128, 1, 1, 1, 9, 85, 663, 3651, 11843, 17060, 6435, 256, 1
Offset: 0
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 4, 10, 19, 31, 46, 64, ...
1, 8, 35, 98, 213, 396, 663, ...
1, 16, 126, 531, 1556, 3651, 7391, ...
1, 32, 462, 2974, 11843, 35232, 86488, ...
-
a(n, k) = polcoef(1/(2-sum(j=0, n, binomial(k*j+1, j)/(k*j+1)*x^j+x*O(x^n))), n);
-
a(n, k) = if(n==0, 1, (binomial(k*n, n)-(k-2)*sum(j=0, n-1, binomial(k*n, j)))/2);
Showing 1-2 of 2 results.
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