OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
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)) ) ) ) ).
EXAMPLE
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.
MAPLE
b:= proc(n, i, p) option remember; `if`(i<1, 0,
`if`(irem(n, i)=0, (j-> (p+j)!/j!*(p+j+1)/2*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
PROG
(PARI)
A_xy(N) = {my(x='x+O('x^N), h = sum(m=1, N, sum(i=1, N, ((y^i)*x^m)*((x^(m+1))/(1-x))^(i-1)*(sum(j=0, N-m-i, prod(u=1, j, (x^(m+1))/(1-x)+(y^(u+i))*x^m)))))); h}
P_xy(N) = Pol(A_xy(N), {x})
A_x(N) = {my(px = deriv(P_xy(N), y), y=1); Vecrev(eval(px))}
A_x(20)
CROSSREFS
KEYWORD
nonn,easy,new
AUTHOR
John Tyler Rascoe, Nov 08 2024
STATUS
approved