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A373322
The number of indecomposable summands, counted with multiplicity, in tensor powers of the vector representation of SL2 in characteristic 2.
0
1, 1, 1, 3, 3, 9, 9, 29, 29, 99, 99, 351, 351, 1273, 1273, 4679, 4679, 17341, 17341, 64637, 64637, 242019, 242019, 909789, 909789, 3432751, 3432751, 12998311, 12998311, 49387289, 49387289, 188261329, 188261329, 719860679, 719860679, 2760525963, 2760525963, 10614508493, 10614508493
OFFSET
0,4
COMMENTS
In characteristic zero the analogous numbers are A001405.
LINKS
K. Coulembier, P. Etingof, V. Ostrik, and D. Tubbenhauer, Fractal behavior of tensor powers of the two dimensional space in prime characteristic, arXiv:2405.16786 [math.RT], 2024.
Daniel Tubbenhauer, GitHub page
FORMULA
a(0) = a(1) = 1, and for n>1: a(2n-1) = a(2n) = Sum_{k=0..n-1} binomial(n-1,k)*2^(n-1-k)*a(k).
a(n) ~ h(n)*n^(t)*2^n for t=1/2log_2(3/2)-1 approx. -0.707 and h(n) a bounded function. The constant t is A153460 - 2.
MATHEMATICA
a[0|1] = 1; a[n_] := a[n] = With[{m = Ceiling[n/2]}, Sum[Binomial[m-1, k] 2^(m-1-k) a[k], {k, 0, m-1}]]; Table[a[n], {n, 0, 40}]
PROG
(PARI) a(n) = if (n<=1, 1, my(m=ceil(n/2)); sum(k=0, m-1, binomial(m-1, k)*2^(m-1-k)*a(k))); \\ Michel Marcus, Jun 01 2024
CROSSREFS
Cf. A001405 (for characteristic zero), A153460.
Sequence in context: A146788 A147244 A146575 * A100066 A170832 A232281
KEYWORD
nonn
AUTHOR
Daniel Tubbenhauer, Jun 01 2024
STATUS
approved