Inclusive bottomonium hadroproduction at the Tevatron is firstly examined in a Monte Carlo framework with the colour-octet mechanism implemented in the event generation. We extract some NRQCD colour-octet matrix elements relevant for $Upsilon(1S)$ hadroproduction. Remarkably we find a quite small contribution (compatible with zero) from feeddown of $chi_{bJ}$ states produced through the colour-octet mechanism: $Upsilon(1S)$ indirect production via $chi_{bJ}$ decays should be mainly ascribed to the colour-singlet model. Finally we extrapolate to LHC energies to predict prompt $Upsilon(1S)$ production rates.
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