A269499 Nontrivial integer solutions s to the equations Sum_{i} ((-1)^i)*binomial(m,i)*binomial(s-m,t-i) = 0 listed in increasing order.

36, 66, 67, 98, 132, 177, 214, 289, 345, 465, 514, 576, 774, 932, 1029, 1219, 1252

Offset: 1

Author

René Gy, Feb 28 2016

Keywords

• nonn
• more

Comments

A nontrivial integer solution s to the equations S(m,s,t) = Sum_{i} (((-1)^i)*binomial(m, i)*binomial(s - m, t - i)) = 0 is an integer s such that there exist integers m, t and 3 < m,t < s/2 such that S(m,s,t)=0 and m,s,t are not such that s=8*k+1 and t=2*k or m=2*k for some integer k.

S(m,s,t)=0 iff S(t,s,m)=0 iff S(s-m,s,t)=0 iff S(s-t,s,m)=0.

For t(resp. m)=2, s=(k+2)^2, m(resp. t)=((k + 2)*(k + 1))/2 is an infinite family of trivial solutions.

For t(resp. m)=3, s=3*k^2 + 8*k + 6, m(resp. t)=((k + 1)*(3*k + 2))/2 is another infinite family of trivial solutions.

For t(resp. m)=3, s=3*k^2 + 10*k + 9, m(resp. t)=((k + 1)*(3*k + 4))/2 is another infinite family of trivial solutions.

For t(resp. m)=2*k, s=8*k+1, m(resp. t)=4*k-1 is another infinite family of trivial solutions.

1521, 3193, 3362, 10882, 15043, 19600 also belong to the sequence, but the list has been checked to be complete only up to 1252.

A082639(k) (from k=4) is included in the sequence, because Sum_{i} (((-1)^i)*binomial(m(k), i)*binomial(s(k) - m, t(k) - i)) = 0, with s(k) = A082639(k) and m(k) = (g^k + g^(-k) - 10)/4 with g=3+2*sqrt(2) and t(k) = (h*g^k + 2 h^(-1)*g^(-k) - 4)/8 with h=2-2*sqrt(2). In other words, they are positive integers s of the form s=2*m+4 where (m,t) m>6 is any couple of positive integer solutions to the Diophantine equation m^2 - 4*m*t + 2*t^2 + 3*m - 8*t + 2 = 0 (there are infinitely many).

Nine such infinite subsequences for the present sequence are known.

The eight other similar subsequences are:

- positive integers s of the form s=2*m+5 where (m,t) is a solution to the Diophantine equation 5*m^2 - 10*m*t + 4*t^2 + 25*m - 26*t + 32 = 0 (producing 2 subsequences).

- positive integers s of the form s=2*m+5 where (m,t) m>10 is a solution to the Diophantine equation m^2 - 6*m*t + 4*t^2 + 3*m - 14*t + 2 = 0 (producing 2 subsequences).

- positive integers s of the form s=2*m+6 where (m,t) is a solution to the Diophantine equation m^2 - 8*m*t + 4*t^2 + 3*m - 24*t + 2 = 0 (producing 2 subsequences).

- positive integers s of the form s=2*m+8 where (m,t) m>7 is a solution to the Diophantine equation m^2 - 4*m*t + 2*t^2 + 7*m - 16*t + 16 = 0 (producing 2 subsequences).

A finite number of nontrivial (sporadic) terms of the sequence (not belonging to one of the above nine subsequences) are also known: 67, 289, 345, 1029, 1521, 10882, 15043, 48324.

Examples

			36=14+22 belongs to the sequence because Sum_{i=0..5} (((-1)^i)*binomial(14, i)*binomial(22,5-i)) = 0, both 5 and 14 are less than 18 and (14,36,5) is not in one of the above trivial families.
		

Links

• René Gy, Trying to solve the equation Sum_{i}(-1)^i*binomial(m,i)*binomial(n-m,t-i)=0 for non-negative integers m,n,t, Math StackExchange.
• L. Habsieger and D. Stanton, More Zeros of Krawtchouk Polynomials, IMA Preprint Series #441, August 1988.
• I. Krasikov, On Integral Zeros of Krawtchouk Polynomials, Journal of Combinatorial Theory, Series A 74, 71-99 (1996).

Crossrefs

Cf. A269563, A082639.

Programs

Mathematica

f[n_,m_,t_]:= Sum[(-1)^i*Binomial[m, i]*Binomial[n-m,t-i],{i,0,t}];lim=200; list={}; Do[ Do[Do [If[ Mod[n,8]==1&& t==2*Quotient[n,8],Continue,If[f[n,m,t]==0  ,AppendTo[list,n]]],{t,4,m}] ,{m,4,n/2-1}],{n,10,lim}];Print [Union [list]]

PARI

isok(s) = {for (m=4, s\2-1, for (t=4, m, if (!(((s % 8) == 1) && (t == 2*(s\8))), if (sum(i=0, t, (-1)^i*binomial(m, i)*binomial(s-m, t-i)) == 0, return (1)););););} \\ Michel Marcus, Mar 01 2016

Extensions

a(16)-a(17) from Michel Marcus, Apr 04 2016