%I #20 Nov 14 2024 08:20:41

%S 0,1,4,10,29,70,181,435,1046,2470,5762,13283,30371,68847,154935,

%T 346433,770154,1703152,3748574,8214805,17931172,38997819,84531066,

%U 182661514,393578129,845777569,1813017039,3877390908,8274351482,17621535902,37456091552,79472869966

%N Sum of the positions of minimum parts in all compositions of n.

%H Alois P. Heinz, <a href="/A377824/b377824.txt">Table of n, a(n) for n = 0..1000</a>

%F 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)) ) ) ) ).

%e 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.

%p b:= proc(n, i, p) option remember; `if`(i<1, 0,

%p `if`(irem(n, i)=0, (j-> (p+j)!/j!*(p+j+1)/2*j)(n/i), 0)+

%p add(b(n-i*j, i-1, p+j)/j!, j=0..(n-1)/i))

%p end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..31); # _Alois P. Heinz_, Nov 12 2024

%o (PARI)

%o 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}

%o P_xy(N) = Pol(A_xy(N), {x})

%o A_x(N) = {my(px = deriv(P_xy(N),y), y=1); Vecrev(eval(px))}

%o A_x(20)

%Y Cf. A001792, A010054, A011782, A097976, A097979.

%K nonn,easy,new

%O 0,3

%A _John Tyler Rascoe_, Nov 08 2024