A087422 - OEIS

%I #18 Jul 07 2023 14:57:58

%S 1,1,2,8,55,567,7958,142396,3104160,79813513

%N Number of terms in the expansion of Product(x_i + x_{i+1} + ... + x_j) over 1 <= i < j <= n.

%C The sum of the coefficients in the expansion of this product are the superfactorials A000178. - _Robert G. Wilson v_, Aug 02 2005

%e a(3) = 8 because the expansion of (x+y)(y+z)(x+y+z) = x^2y + x^2z + 2xy^2 + 3xyz + xz^2 + y^3 + 2y^2z + yz^2 has 8 terms.

%p a:= n-> nops(expand(mul(mul(add(x[k], k=i..j), i=1..j-1), j=2..n))):

%p seq(a(n), n=0..8);  # _Alois P. Heinz_, Jul 07 2023

%t f[1]=1; f[n_] := Block[{lst = Take[{a, b, c, d, e, f, g, h, i}, n], s = 1}, Do[s = s*Times @@ Plus @@@ Partition[lst, i, 1], {i, 2, n}]; Length@Expand@s]; Do[ Print@ f@n, {n, 9}] (* _Robert G. Wilson v_, Sep 18 2006 *)

%Y Cf. A000178.

%K nonn,more

%O 0,3

%A Alex Postnikov (apost(AT)math.mit.edu), Oct 22 2003

%E a(8) from _Robert G. Wilson v_, Apr 26 2005

%E a(9) from _Vaclav Kotesovec_, Apr 09 2021

%E a(0)=1 prepended by _Alois P. Heinz_, Jul 07 2023