OFFSET
1,3
COMMENTS
All rows are linear recurrences with constant coefficients. See PARI script to obtain generating functions.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.
FORMULA
T(m, n) = Sum_{j=1..m} (1 + 2*cos(j*pi/(m+1)))^n. - Andrew Howroyd, Apr 15 2017
EXAMPLE
Array starts:
1 1 1 1 1 1 1 1 1 1 ...
2 4 8 16 32 64 128 256 512 1024 ...
3 7 15 35 83 199 479 1155 2787 6727 ...
4 10 22 54 134 340 872 2254 5854 15250 ...
5 13 29 73 185 481 1265 3361 8993 24193 ...
6 16 36 92 236 622 1658 4468 12132 33146 ...
7 19 43 111 287 763 2051 5575 15271 42099 ...
8 22 50 130 338 904 2444 6682 18410 51052 ...
9 25 57 149 389 1045 2837 7789 21549 60005 ...
10 28 64 168 440 1186 3230 8896 24688 68958 ...
MATHEMATICA
T[m_, n_] := Sum[(1 + 2*Cos[j*Pi/(m+1)])^n, {j, 1, m}] // FullSimplify;
Table[T[m-n+1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Jun 06 2017 *)
PROG
(PARI) \\ from Knopfmacher et al.
ChebyshevU(n, x) = sum(i=0, n/2, 2*poltchebi(n-2*i, x)) + (n%2-1);
RowGf(k, x) = 1 + (k*x*(1+3*x) - 2*(k+1)*x*subst(ChebyshevU(k-1, z)/ChebyshevU(k, z), z, (1-x)/(2*x)))/((1+x)*(1-3*x));
a(m, n)=Vec(RowGf(m, x)+O(x^(n+1)))[n+1];
for(m=1, 10, print(RowGf(m, x)));
for(m=1, 10, for(n=1, 9, print1( a(m, n), ", ") ); print(); );
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Apr 15 2017
STATUS
approved