cp's OEIS Frontend

A119551 Number of ordered finite sequences a_1 <= a_2 <= ... <= a_n of length n of positive integers less than or equal to n whose product is n! and whose sum is n * (n + 1) / 2.

%I A119551 #28 Feb 16 2023 05:19:31
%S A119551 1,1,1,1,1,1,1,1,1,2,6,6,22,22,60,159,377,377,1007,1007,2867,8147,
%T A119551 22403,22403,67808,176128,495053,1362240,4210266,4210266,14223808,
%U A119551 14223808,42235255,129279396,370630653,1178215490
%N A119551 Number of ordered finite sequences a_1 <= a_2 <= ... <= a_n of length n of positive integers less than or equal to n whose product is n! and whose sum is n * (n + 1) / 2.
%C A119551 a(n) is also the number of lattice points in a sequence of polytopes. Given n, define a vector x(k) = #{j : a_j = k} and define a matrix A with n columns as follows: first row all 1 (gives length of a_j); second row 1,2,...,n (sum of a_j); finally one row for each prime p <= n with entries A(row p, column k) = maximum exponent of p that divides k, e.g., A(p=2,k=8)=3 because 2^3|8 (this gives factorization of product of a_j).  Then a(n) is the number of nonnegative integer lattice points in the polytope A*x = A*(1,1,1...)T. - _Martin Fuller_, Feb 12 2023
%H A119551 Martin Fuller, <a href="/A119551/b119551.txt">Table of n, a(n) for n = 0..61</a>
%F A119551 a(p) = a(p-1) for prime p. - _Alois P. Heinz_, Jul 05 2022
%e A119551 a(9) = 2 because the sequences (1, 2, 3, 4, 5, 6, 7, 8, 9) and (1, 2, 4, 4, 4, 5, 7, 9, 9) both add up to 45 and multiply up to 9!.
%t A119551 a[n_] := a[n] = Module[{b}, b[c_, s_, p_, m_] := b[c, s, p, m] = Module[{x}, If[c <= 0 || m <= 1 || s <= c || s > m*c, Boole[ c == s && p == 1], x = IntegerExponent[p, m]; Sum[b[c - i, s - m*i, p/m^i, m - 1], {i, x*Boole@PrimeQ[m], x} ]]]; b[n, n*(n + 1)/2, n!, n]];
%t A119551 Table[Print[n, " ", a[n]]; a[n], {n, 1, 32}] (* _Jean-François Alcover_, Jul 05 2022, after _Martin Fuller_ *)
%o A119551 (PARI) a(n) = (b(c,s,p,m) = local(x); if(c<=0||m<=1||s<=c||s>m*c, c==s&&p==1, x=valuation(p,m); sum(i=x*isprime(m), x, b(c-i,s-m*i,p/m^i,m-1)))); b(n,n*(n+1)/2,n!,n) \\ _Martin Fuller_, Jun 26 2006
%Y A119551 Cf. A000040, A000142, A000217, A076822 without restriction on product, A120690 without restriction on sum.
%K A119551 nonn,nice
%O A119551 0,10
%A A119551 _Jens Voß_, May 30 2006
%E A119551 a(18) and a(19) from _John W. Layman_, Jun 08 2006
%E A119551 More terms from _Martin Fuller_, Jun 26 2006
%E A119551 a(0)=1 prepended by _Alois P. Heinz_, Jul 05 2022
%E A119551 a(36)-a(61) from _Martin Fuller_, Feb 12 2023