A377823 Sum of the positions of maximum parts in all compositions of n.

0, 1, 4, 10, 23, 50, 110, 240, 526, 1147, 2489, 5368, 11510, 24543, 52090, 110109, 231959, 487245, 1020980, 2134838, 4455582, 9283742, 19314740, 40128699, 83265342, 172564435, 357228078, 738707908, 1526004117, 3149310585, 6493394292, 13376521031, 27532616663

Offset: 0

Author

John Tyler Rascoe, Nov 08 2024

Keywords

• nonn
• easy

Examples

			The composition of 7, (1,2,1,1,2) has maximum parts at positions 2 and 5; so it contributes 7 to a(7) = 240.
		

Crossrefs

Cf. A001792, A010054, A011782, A097976, A097979, A377824.

Programs

PARI

A_xy(N) = {my(x='x+O('x^N), h = sum(i=1,N, y^(i*(i+1)/2)*x^i)+sum(m=2,N, sum(i=1,N, ((y^i)*x^m)*((x-x^m)/(1-x))^(i-1)*(sum(j=0,N-m-i, prod(u=1,j, (x-x^m)/(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)

Formula

G.f.: A(x) = d/dy A(x,y)|{y = 1}, where A(x,y) = Sum{i>0} (x^i * y^(i*(i+1)/2)) + Sum_{m>1} (Sum_{i>0} (x^m * y^i * ((x-x^m)/(1-x))^(i-1) * (Sum_{j>=0} (Product_{u=1..j} ((x-x^m)/(1-x) + x^m * y^(u+i)) ) ) ) ).