Is there a formula expressing the coefficients of the product ∏ k = 1 n ( ∑ j = 0 ∞ ( j + 1 ) x j k ) ,

Question

Is there a formula expressing the coefficients of the product      ∏          k      =      1        n        (          ∑              j        =        0            ∞        (    j    +    1    )          x              j        k              )    ,

Nathen Austin · Accepted Answer

Let      f    n    (  x  )  =      ∏          k      =      1        n    (  1  −      x    k        )          −      1      It is well known that      f    n    (  x  )  =      ∑          j      ≥      0            p          ≤      n        (  j  )      x    j  where       p    n    (  j  ) is the number of partitions of j into parts whose sizes are all at most n. Equivalently, via conjugation,       p    n    (  j  ) is the number of partitions of j into parts at most n parts.Your generating function is      ∏          k      =      1        n        (          ∑              j        =        0            ∞        (    j    +    1    )          x              j        k              )    =      ∏          k      =      1        n    (  1  −      x    k        )          −      2        =  [      f    n    (  x  )      ]    2  Therefore, since the product of generating functions leads to a convolution of coefficient sequences, we concludecoefficient of       coefficient of                   x        m               in           [              f        n            (      x      )              ]        2              =      p          ≤      n        (  0  )      p          ≤      n        (  m  )  +      p          ≤      n        (  1  )      p          ≤      n        (  m  −  1  )  +  ⋯  +      p          ≤      n        (  m  )      p          ≤      n        (  0  )

Is there a formula expressing the coefficients of the product <munderover> ∏<!-- ∏ --

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