A377824 Sum of the positions of minimum parts in all compositions of n.
0, 1, 4, 10, 29, 70, 181, 435, 1046, 2470, 5762, 13283, 30371, 68847, 154935, 346433, 770154, 1703152, 3748574, 8214805, 17931172, 38997819, 84531066, 182661514, 393578129, 845777569, 1813017039, 3877390908, 8274351482, 17621535902, 37456091552, 79472869966
Offset: 0
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
Programs
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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 -
Mathematica
b[n_, i_, p_] := b[n, i, p] = If[i < 1, 0, If[Mod[n, i] == 0, Function[j, (p + j)!/j!*(p + j + 1)/2*j][n/i], 0] + Sum[b[n - i*j, i - 1, p + j]/j!, {j, 0, (n - 1)/i}]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 31}] (* Jean-François Alcover, Apr 19 2025, after Alois P. Heinz *) -
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)
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)) ) ) ) ).
Conjecture: a(n) ~ n^2 * 2^(n-5). - Vaclav Kotesovec, Apr 19 2025