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
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.
Programs
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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)) ) ) ) ).