1,3

The a(n,p) polynomials, see below with the extra p for the column number, generate the coefficients of the left hand columns of triangle A163940. These polynomials are interesting in their own right. They have many curious properties e.g. for p=>1: a(n=1, p) = p, a(n=0, p) = 0, a(n = -1, p) = (-1)^(p+1), a(n=-2,p) = (-1)^(p+1)*(2)^(p-2) and a(n = -(2*p+1), 2*p) = 0, which is the outermost zero of the a(n, 2*p); for p=>10: a(n=-10, p) = -362880*10^(p-10); etc..

The numbers in the denominators of the a(n,p) are the Minkowski numbers A053657.

The Maple program generates the coefficients of the polynomials that appear in the numerators of the a(n,p), see the sequence above. We have made use of a nice little program that Peter Luschny recently wrote for the Minkowski numbers! For the an(p,k) in the Maple program for p =>1 we have 0 <= k <= (2*p-2). A word of caution.The value of nmax has to be chosen sufficiently large in order to let Maple find the o.g.f.s..

The zero patterns of the a(n,p) polynomials resemble the Montezuma Cypress (Taxodium mucronatum). A famous Montezuma Cypress is 'El Arbol del Tule' (the Tule tree) in Mexico. It is the second stoutest tree in the world, circumference 36 meter, and is approximately 1500 years old. Considering this I propose to call the a(n,p) polynomials the MC polynomials.

The a(n,p) formulae of the first few left hand columns of the A163940 triangle (p is the column number):

a(n,1) = (1)/1

a(n,2) = (0 + 3*n + n^2)/2

a(n,3) = (0 + 2*n + 45*n^2+ 22*n^3 + 3*n^4)/24

a(n,4) = (0 + 0*n + 10*n^2 + 107*n^3 + 61*n^4 + 13*n^5 + n^6)/48

a(n,5) = (0 - 48*n + 20*n^2 + 2100*n^3 + 14855*n^4 + 9168*n^5 + 2390*n^6 + 300*n^7 + 15*n^8)/5760

a(n,6) = (0 + 0*n -336*n^2 +92*n^3 +6320*n^4 +33765*n^5 +21803*n^6 +6378*n^7 +1010*n^8 +85*n^9 +3*n^10)/11520

a(n,7) = (0 + 11520*n -2016*n^2 -198296*n^3 +33012*n^4 +2199246*n^5 +9547461*n^6+ 6331782*n^7 +1994265*n^8 +362474*n^9 +39375*n^10 +2394*n^11 +63*n^12)/2903040

pmax:=6; nmax:=70; with(genfunc): A053657 := proc(n) local P, p, q, s, r; P := select(isprime, [$2..n]); r:=1; for p in P do s := 0: q := p-1; do if q > (n-1) then break fi; s := s + iquo(n-1, q); q := q*p; od; r := r * p^s; od; r end: for px from 1 to nmax do Gf(px):= convert(series(1/((1-(px-1)*x)^2*product((1-k*x), k=1..px-2)), x, nmax+1-px), polynom): for qy from 0 to nmax-px do a(px+qy, qy):=coeff(Gf(px), x, qy) od; od: for p from 1 to pmax do f(x):=0: for ny from p to nmax do f(x):=f(x)+a(ny, p-1)*x^(ny-p) od: f(x):= series(f(x), x, nmax): Gx:=convert(%, ratpoly): rgf_sequence('recur', Gx, x, G, n): a(n, p):=sort(simplify (rgf_expand(Gx, x, n)), n): f(p):=sort(a(n, p)*A053657(p), n, ascending): for k from 0 to 2*p-2 do an(p, k):= coeff(f(p), n, k) od; od: T:=1: for p from 1 to pmax do for k from 0 to 2*p-2 do a(T):=an(p, k): T:=T+1 od: od: seq(a(n), n=1..T-1); for p from 1 to pmax do seq(an(p, k), k=0..2*p-2) od; for p from 1 to pmax do MC(n, p):=sort(a(n, p), n, ascending) od;

A000012, A000096, A163943 and A163944 are the first four left hand columns of A163940.

Cf. A053657 (Minkowski), A163402 and A075264.

Sequence in context: A073200 A104416 A144108 this_sequence A068464 A135297 A035674

Adjacent sequences: A163969 A163970 A163971 this_sequence A163973 A163974 A163975

easy,sign,tabf

Johannes W. Meijer (meijgia(AT)hotmail.com), Aug 13 2009

Edited by Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 06 2009