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Triangle read by rows where row m is the m-th Gilbreath polynomial and column n is the numerator of the coefficient of the n-th degree term.
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%I #17 Oct 25 2021 14:32:55

%S 1,1,0,1,0,0,-1,-3,-1,0,-5,-5,-1,0,0,-57,-55,-15,-2,0,0,-282,-232,-77,

%T -14,-1,0,0,-3270,-2554,-850,-175,-20,-1,0,0,-41940,-30948,-10654,

%U -2325,-325,-27,-1,0,0,-608580,-437772,-152180,-34174,-5285,-553,-35,-1,0,0

%N Triangle read by rows where row m is the m-th Gilbreath polynomial and column n is the numerator of the coefficient of the n-th degree term.

%C Let S=(p_1, ..., p_m) be the ordered sequence of the first m prime numbers. The m-th Gilbreath polynomial is defined as the polynomial P_m such that the x-th term of the upper bound Gilbreath sequence of S, U(S)_x, is U(S)_x = 2^(m+x-1) + P_m where P_m = Sum_{n=1..m} T(m,n)*x^(n-1)/A347925(m).

%C The values T(m,1), ..., T(m,n) are the numerators of the coefficients of the (n-th)-degree terms of the m-th Gilbreath polynomial.

%H Riccardo Gatti, <a href="https://www.preprints.org/manuscript/202003.0145">Gilbreath Sequences and Proof of Conditions for Gilbreath Conjecture</a>, Preprints 2020, 2020030145.

%H Riccardo Gatti, <a href="https://github.com/gttrcr/ResearchCode/blob/main/OEIS/A347924.cs">Program for the generation of the m-th Gilbreath polynomial calling GenMthGilbreathPolynomial(m)</a>

%H A. M. Odlyzko, <a href="https://doi.org/10.1090/S0025-5718-1993-1182247-7">Iterated absolute values of differences of consecutive primes</a>, Math. Comp. 61 (1993), 373-380.

%e Consider the triangle T(m,n) of the first terms of the sequence:

%e m\n 1 2 3 4 5 6 7 8

%e 1 1

%e 2 1 0

%e 3 1 0 0

%e 4 -1 -3 -1 0

%e 5 -5 -5 -1 0 0

%e 6 -57 -55 -15 -2 0 0

%e 7 -282 -232 -77 -14 -1 0 0

%e ...

%e The terms associated to P_6 are -57, -55, -15, -2, 0, 0. The numerators of coefficients of P_6 are in order of degree of the term of the polynomial: -57 for the term of degree 0, -55 for the term of degree 1 and so on until 0 for the terms of degree 4 and 5. Hence P_6 = (-57 - 55x - 15x^2 - 2x^3)/3, where A347925(6)=3, in fact the x-th term of the upper bound Gilbreath sequence of S=(p_1, ..., p_6) = (2, 3, 5, 7, 11, 13) is U(S)_x = 2^(x+5) + (-57 - 55x - 15x^2 - 2x^3)/3.

%o (PARI) polynomialfit(data) = Pol(Vecrev(matsolve(matrix(#data,#data,i,j,i^(j-1)), data~))); \\ from _David A. Corneth_

%o isg(v, k) = {my(w = concat(v, k), vd = w); for (i=1, #w-1, vd = vector(#vd-1, k, abs(vd[k+1] - vd[k])); if (vd[1] != 1, return (0));); return (1);}

%o nextx(v) = {my(k = nextprime(nextprime(vecmax(v)+1)+1)); while (isg(v, k), k+=2); k-=2;}

%o row(n) = {my(vp = primes(n), v = List()); for (i=1, n, my(x = nextx(vp)); vp = concat(vp, x); listput(v, x);); v = Vec(v); my(cp = Vecrev(polynomialfit(vector(#v, k, v[k] - 2^(k+n-1))))); my(k = lcm(apply(denominator, cp))); while(#cp != n, cp = concat(cp, 0)); cp *= k;} \\ _Michel Marcus_, Sep 20 2021

%Y Cf. A347925.

%K sign,frac,tabl

%O 1,8

%A _Riccardo Gatti_, Sep 20 2021