A373322 The number of indecomposable summands, counted with multiplicity, in tensor powers of the vector representation of SL2 in characteristic 2.

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

Author

Daniel Tubbenhauer, Jun 01 2024

Keywords

• nonn

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.
• M. Larsen, Bounds for SL2-indecomposables in tensor powers of the natural representation in characteristic 2, arXiv:2405.16015 [math.RT], 2024.
• Daniel Tubbenhauer, GitHub page

Crossrefs

Cf. A001405 (for characteristic zero), A153460.

Programs

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}]

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

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.