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A119551
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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.
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2
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1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 6, 6, 22, 22, 60, 159, 377, 377, 1007, 1007, 2867, 8147, 22403, 22403, 67808, 176128, 495053, 1362240, 4210266, 4210266, 14223808, 14223808, 42235255, 129279396, 370630653, 1178215490
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OFFSET
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0,10
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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!.
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MATHEMATICA
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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]];
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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