A373536 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.

1, 1, 9, 364, 61320, 41747328, 113420740608, 1223445790457856, 52307167449899335680, 8861896666997422628536320, 5951934931285476447488997064704, 15857359709817958217841735837828513792, 167702614892018104786663957623269078052372480, 7044769706183185876455816992603242619680927682396160

Offset: 0

Author

Geoffrey Critzer, Jun 08 2024

Keywords

• nonn

Links

• David Ellerman, The number of direct-sum decompositions of a finite vector space, arXiv:1603.07619 [math.CO], 2016.
• R. P. Stanley, Exponential structures

Crossrefs

Cf. A270880, A000262, A053601, A002884.

Programs

Mathematica

nn = 13; B[n_] := Product[q^n - q^i, {i, 0, n - 1}] /. q -> 2;
e[x_] := Sum[x^n/B[n], {n, 0, nn}]; f[x_] := Sum[x^n, {n, 0, nn}];
Table[B[n], {n, 0, nn}] CoefficientList[Series[Exp[f[x] - 1], {x, 0, nn}], x]

Formula

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

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