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%I #78 Jan 21 2023 04:55:14
%S 36,66,67,98,132,177,214,289,345,465,514,576,774,932,1029,1219,1252
%N Nontrivial integer solutions s to the equations Sum_{i} ((-1)^i)*binomial(m,i)*binomial(s-m,t-i) = 0 listed in increasing order.
%C 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.
%C 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.
%C For t(resp. m)=2, s=(k+2)^2, m(resp. t)=((k + 2)*(k + 1))/2 is an infinite family of trivial solutions.
%C 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.
%C 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.
%C For t(resp. m)=2*k, s=8*k+1, m(resp. t)=4*k-1 is another infinite family of trivial solutions.
%C 1521, 3193, 3362, 10882, 15043, 19600 also belong to the sequence, but the list has been checked to be complete only up to 1252.
%C 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).
%C Nine such infinite subsequences for the present sequence are known.
%C The eight other similar subsequences are:
%C - 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).
%C - 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).
%C - 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).
%C - 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).
%C 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.
%H René Gy, <a href="https://math.stackexchange.com/q/1664420/130022"> 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</a>, Math StackExchange.
%H L. Habsieger and D. Stanton, <a href="https://www.ima.umn.edu/preprints/More-Zeros-Krawtchouk-Polynomials">More Zeros of Krawtchouk Polynomials</a>, IMA Preprint Series #441, August 1988.
%H I. Krasikov, <a href="https://doi.org/10.1006/jcta.1996.0038">On Integral Zeros of Krawtchouk Polynomials</a>, Journal of Combinatorial Theory, Series A 74, 71-99 (1996).
%e 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.
%t 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]]
%o (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
%Y Cf. A269563, A082639.
%K nonn,more
%O 1,1
%A _René Gy_, Feb 28 2016
%E a(16)-a(17) from _Michel Marcus_, Apr 04 2016
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