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

0,3

Links

Table of n, a(n) for n=0..32.

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)) ) ) ) ).

Example

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.

Prog

(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)

Crossrefs

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

Sequence in context: A001980 A266376 A057750 * A295059 A118645 A200759

Adjacent sequences: A377819 A377820 A377821 * A377824 A377825 A377826

Keyword

nonn,easy,changed

Author

John Tyler Rascoe, Nov 08 2024

Status

approved