Find the generating function for the sequence an, where an is the number of natural solutions to the equation x1 + x2 + 2x3 + 5x4 = n with xi 0 for i...
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1.Find the generating function for the sequence an, where an is the number of natural solutions to the equation x1

+ x2 + 2x3 + 5x4 = n with xi ≥ 0 for i = 1, 2, 3, 4.

2.Show that the number of partitions of 2n into distinct even parts isequal to the number of partitions of n into odd parts.

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1.
x1 + x2 + 2x3 + 5x4 = n
The generating function of the sequence is given by
x1 + (x2) X + (2x3) X^2 + (5x4) X^3
¿
xi¿ =x1 + 4 ∑ i=2,n=1 2. ¿ 2+2n) Xn This shows that the number of...

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