A373536 - OEIS

%I #8 Jun 28 2024 22:57:15

%S 1,1,9,364,61320,41747328,113420740608,1223445790457856,

%T 52307167449899335680,8861896666997422628536320,

%U 5951934931285476447488997064704,15857359709817958217841735837828513792,167702614892018104786663957623269078052372480,7044769706183185876455816992603242619680927682396160

%N Number of ways to form a direct sum decomposition of the vector space GF(2)^n and then choose a basis for each subspace in the decomposition.

%H David Ellerman, <a href="http://arxiv.org/abs/1603.07619">The number of direct-sum decompositions of a finite vector space</a>, arXiv:1603.07619 [math.CO], 2016.

%H R. P. Stanley, <a href="http://www-math.mit.edu/~rstan/pubs/pubfiles/37.pdf">Exponential structures</a>

%F a(n) = A000262(n)*A053601(n).

%F Sum_{n>=0} a(n)*x^n/A002884(n) = exp(x/(1-x)).

%t nn = 13; B[n_] := Product[q^n - q^i, {i, 0, n - 1}] /. q -> 2;

%t e[x_] := Sum[x^n/B[n], {n, 0, nn}]; f[x_] := Sum[x^n, {n, 0, nn}];

%t Table[B[n], {n, 0, nn}] CoefficientList[Series[Exp[f[x] - 1], {x, 0, nn}], x]

%Y Cf. A270880, A000262, A053601, A002884.

%K nonn

%O 0,3

%A _Geoffrey Critzer_, Jun 08 2024