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pentagonal numbers

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1: 27.14 Unrestricted Partitions
Euler’s pentagonal number theorem states that
27.14.4 f ( x ) = 1 - x - x 2 + x 5 + x 7 - x 12 - x 15 + = 1 + k = 1 ( - 1 ) k ( x ω ( k ) + x ω ( - k ) ) ,
where the exponents 1 , 2 , 5 , 7 , 12 , 15 , are the pentagonal numbers, defined by
27.14.5 ω ( ± k ) = ( 3 k 2 k ) / 2 , k = 1 , 2 , 3 , .
27.14.6 p ( n ) = k = 1 ( - 1 ) k + 1 ( p ( n - ω ( k ) ) + p ( n - ω ( - k ) ) ) = p ( n - 1 ) + p ( n - 2 ) - p ( n - 5 ) - p ( n - 7 ) + ,
2: 20.12 Mathematical Applications
For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s τ ( n ) function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …
3: Bibliography
  • G. E. Andrews (2000) Umbral calculus, Bailey chains, and pentagonal number theorems. J. Combin. Theory Ser. A 91 (1-2), pp. 464–475.