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ON THE DIVISIBILITY OF SUMS OF q-SUPER CATALAN NUMBERS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 109 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 15 May 2023, pp. 215-224
- Print publication:
- April 2024
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- Article
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The integrality of the numbers
$A_{n,m}={(2n)!(2m)!}/{n!m!(n+m)!}$ was observed by Catalan as early as 1874 and Gessel named 
$A_{n,m}$ the super Catalan numbers. The positivity of the q-super Catalan numbers (q-analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin [‘A q-rious positivity’, Aequationes Math. 81 (2011), 177–183]. We prove the divisibility of sums of q-super Catalan numbers, which establishes a q-analogue of Apagodu’s congruence involving super Catalan numbers.
