

A269499


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


2



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

1,1


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(sm,s,t)=0 iff S(st,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*k1 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=22*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.


LINKS

Table of n, a(n) for n=1..17.
René Gy, Trying to solve the equation Sum_{i}(1)^i*binomial(m,i)*binomial(nm,ti)=0 for nonnegative 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, 7199 (1996).


EXAMPLE

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


MATHEMATICA

f[n_, m_, t_]:= Sum[(1)^i*Binomial[m, i]*Binomial[nm, ti], {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/21}], {n, 10, lim}]; Print [Union [list]]


PROG

(PARI) isok(s) = {for (m=4, s\21, for (t=4, m, if (!(((s % 8) == 1) && (t == 2*(s\8))), if (sum(i=0, t, (1)^i*binomial(m, i)*binomial(sm, ti)) == 0, return (1)); ); ); ); } \\ Michel Marcus, Mar 01 2016


CROSSREFS

Cf. A269563, A082639.
Sequence in context: A326666 A272190 A060671 * A074315 A240520 A126789
Adjacent sequences: A269496 A269497 A269498 * A269500 A269501 A269502


KEYWORD

nonn,more


AUTHOR

René Gy, Feb 28 2016


EXTENSIONS

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


STATUS

approved



