LINKS
Alois P. Heinz, <a href="/A377824/b377824_1.txt">Table of n, a(n) for n = 0..1000</a>
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Alois P. Heinz, <a href="/A377824/b377824_1.txt">Table of n, a(n) for n = 0..1000</a>
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G.f.: A(x) = d/dy A(x,y)|_{y = 1, }, where A(x,y) = Sum_{m>0} (Sum_{i>0} (x^m * y^i * (x^(m+1)/(1-x))^(i-1) * (Sum_{j>=0} (Product_{u=1..j} (x^(m+1)/(1-x) + x^m * y^(u+i)) ) ) ) ).
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b:= proc(n, i, p) option remember; `if`(i<1, 0, `if`(irem(n, i)=0,
`if`(irem(n, i)=0, (j-> (p+j)!/j!*(p+j)*(p+j+1)/2*j/(p+j))(n/i), 0)+
add(b(n-i*j, i-1, p+j)/j!, j=0..(n-1)/i))
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b:= proc(n, i, p) option remember; `if`(i<1, 0, `if`(irem(n, i)=0,
(j-> (p+j)!/j!*(p+j)*(p+j+1)/2*j/(p+j))(n/i), 0)+
add(b(n-i*j, i-1, p+j)/j!, j=0..(n-1)/i))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..31); # Alois P. Heinz, Nov 12 2024
The composition of 7, (1,2,1,1,2) has minimum parts at positions 1, 3, and 4; so it contributes 8 to a(7) = 435.
The composition of 7, (1, 2, 1, 1, 2) has minimum parts at positions 1, 3, and 4; so it contributes 8 to a(7) = 435.
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