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A377824 Sum of the positions of minimum parts in all compositions of n.

%I A377824 #28 Apr 19 2025 08:48:50
%S A377824 0,1,4,10,29,70,181,435,1046,2470,5762,13283,30371,68847,154935,
%T A377824 346433,770154,1703152,3748574,8214805,17931172,38997819,84531066,
%U A377824 182661514,393578129,845777569,1813017039,3877390908,8274351482,17621535902,37456091552,79472869966
%N A377824 Sum of the positions of minimum parts in all compositions of n.
%H A377824 Alois P. Heinz, <a href="/A377824/b377824.txt">Table of n, a(n) for n = 0..1000</a>
%F A377824 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)) ) ) ) ).
%F A377824 Conjecture: a(n) ~ n^2 * 2^(n-5). - _Vaclav Kotesovec_, Apr 19 2025
%e A377824 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 A377824 b:= proc(n, i, p) option remember; `if`(i<1, 0,
%p A377824       `if`(irem(n, i)=0, (j-> (p+j)!/j!*(p+j+1)/2*j)(n/i), 0)+
%p A377824       add(b(n-i*j, i-1, p+j)/j!, j=0..(n-1)/i))
%p A377824     end:
%p A377824 a:= n-> b(n$2, 0):
%p A377824 seq(a(n), n=0..31);  # _Alois P. Heinz_, Nov 12 2024
%t A377824 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}]];
%t A377824 a[n_] := b[n, n, 0];
%t A377824 Table[a[n], {n, 0, 31}] (* _Jean-François Alcover_, Apr 19 2025, after _Alois P. Heinz_ *)
%o A377824 (PARI)
%o A377824 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 A377824 P_xy(N) = Pol(A_xy(N), {x})
%o A377824 A_x(N) = {my(px = deriv(P_xy(N),y), y=1); Vecrev(eval(px))}
%o A377824 A_x(20)
%Y A377824 Cf. A001792, A010054, A011782, A097976, A097979, A377823.
%K A377824 nonn,easy
%O A377824 0,3
%A A377824 _John Tyler Rascoe_, Nov 08 2024