1,1

This is the problem of the farmer's helper who, when asked to weigh n bags of grain, does so k at a time and reports the resulting binomial(n,k) combined weights with no indication of the k-tuples that produced them. The problem: is can the weights of the bags be recovered?

For k=3 the answer is Yes unless n is one of the four terms of this sequence. For k=2 see A057716.

The old entry with this sequence number was a duplicate of A030109.

The following references also apply to the general case of the problem.

I. N. Baker, Solutions of the functional equation (f(x))^2-f(x^2)=h(x), Canad. Math. Bull., 3 (1960) 113-120.

W. W. Rouse Ball, A Short Account of the History of Mathematics.

E. Bolker, The finite Radon transform, Contemp. Math., 63 (1987) 27-50.

J. Boman, E. Bolker and P. O'Neil, The combinatorial Radon transform modulo the symmetric group, Adv. Appl. Math., 12 (1991) 400-411.

Boman, Jan and Linusson, Svante, Examples of non-uniqueness for the combinatorial Radon transform modulo the symmetric group. Math. Scand. 78 (1996), 207-212.

John A. Ewell, On the determination of sets by sets of sums of fixed order, Canad. J. Math., 20 (1968) 596-611.

B. Gordon, A. S. Fraenkel and E. G. Straus, On the determination of sets by the sets of sums of a certain order, Pacific J. Math., 12 (1962) 187-196.

R. K. Guy, Unsolved Problems in Number Theory, C5.

Ross A. Honsberger, A gem from combinatorics, Bull. ICA, 1 (1991) 56-58.

J. Lambek and L. Moser, On some two way classifications of the integers, Canad. Math. Bull., 2 (1959) 85-89.

B. Liu and X. Zhang, On harmonious labelings of graphs, Ars Combin., 36 (1993) 315-326.

L. Moser, Problem E1248, Amer. Math. Monthly, 64 (1957) 507.

J. Ossowski, On a problem of Galvin, Congressus Numerantium, 96 (1993) 65-74.

D. G. Rogers, A functional equation: solution to Problem 89-19*, SIAM Review, 32 (1990) 684-686.

J. L. Selfridge and E. G. Straus, On the determination of numbers by their sums of a fixed order, Pacific J. Math., 8 (1958) 847-856.

P. Winkler, Mathematical Mind-Benders, Peters, Wellesley, MA, 2007; see p. 27.

For n=27 Boman and Linusson give five examples of which the simplest is {-4,-1^{10},2^{16}} and its negative, where exponents denote repetitions. For n=486 Boman and Linusson give {-7,-4^{56},-1^{231},2^{176},5^{22}} and its negative.

See A057716 for the case k=2.

Sequence in context: A083695 A060170 A097678 this_sequence A083675 A085076 A076711

Adjacent sequences: A074891 A074892 A074893 this_sequence A074895 A074896 A074897

nonn,fini,full

N. J. A. Sloane (njas(AT)research.att.com), based on email from R. K. Guy, Oct 30 2008