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1978 IMO Longlists
2
(x + 2x^2 +...+ nx^n)^2 = a_2x^2 + a_3x^3 +...+ a_{2n}x^{2n}
(x + 2x^2 +...+ nx^n)^2 = a_2x^2 + a_3x^3 +...+ a_{2n}x^{2n}
Source:
October 14, 2010
algebra
generating functions
IMO Longlist 1978
Problem Statement
If
f
(
x
)
=
(
x
+
2
x
2
+
⋯
+
n
x
n
)
2
=
a
2
x
2
+
a
3
x
3
+
⋯
+
a
2
n
x
2
n
,
f(x) = (x + 2x^2 +\cdots+ nx^n)^2 = a_2x^2 + a_3x^3 + \cdots+ a_{2n}x^{2n},
f
(
x
)
=
(
x
+
2
x
2
+
⋯
+
n
x
n
)
2
=
a
2
x
2
+
a
3
x
3
+
⋯
+
a
2
n
x
2
n
,
prove that
a
n
+
1
+
a
n
+
2
+
⋯
+
a
2
n
=
(
n
+
1
2
)
5
n
2
+
5
n
+
2
12
a_{n+1} + a_{n+2} + \cdots + a_{2n} =\dbinom{n + 1}{2}\frac{5n^2 + 5n + 2}{12}
a
n
+
1
+
a
n
+
2
+
⋯
+
a
2
n
=
(
2
n
+
1
)
12
5
n
2
+
5
n
+
2
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