A109626 Consider the array T(n,m) where the n-th row is the sequence of integer coefficients of A(x), where 1<=a(n)<=n, such that A(x)^(1/n) consists entirely of integer coefficients and where m is the (m+1)-th coefficient. This is the antidiagonal read from lower left to upper right.

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 3, 2, 1, 1, 5, 2, 1, 2, 1, 1, 6, 5, 4, 3, 2, 1, 1, 7, 3, 5, 3, 3, 1, 1, 1, 8, 7, 2, 5, 4, 3, 2, 1, 1, 9, 4, 7, 3, 1, 4, 3, 2, 1, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 11, 5, 3, 2, 7, 6, 5, 1, 3, 1, 1, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 1, 13, 6, 11, 10, 9, 4, 1, 3, 5

Offset

1,5

Links

G. C. Greubel, Antidiagonals n = 1..100, flattened

N. Heninger, E. M. Rains, N. J. A. Sloane, On the intgrality of nth roots of generatign functions, J. Comb. Theory A 113 (2006) 1732-1745

Formula

When m is prime, column m is T(n,m) = n/gcd(m, n) = numerator of n/(n+m). - M. F. Hasler, Jan 27 2025

Example

Table begins:
\k...0...1...2...3...4...5...6...7...8...9..10..11..12..13
n\
 1|  1   1   1   1   1   1   1   1   1   1   1   1   1   1
 2|  1   2   1   2   2   2   1   2   2   2   1   2   1   2
 3|  1   3   3   1   3   3   3   3   3   3   3   3   1   3
 4|  1   4   2   4   3   4   4   4   1   4   4   4   3   4
 5|  1   5   5   5   5   1   5   5   5   5   4   5   5   5
 6|  1   6   3   2   3   6   6   6   3   4   6   6   6   6
 7|  1   7   7   7   7   7   7   1   7   7   7   7   7   7
 8|  1   8   4   8   2   8   4   8   7   8   8   8   4   8
 9|  1   9   9   3   9   9   3   9   9   1   9   9   6   9
10|  1  10   5  10  10   2   5  10  10  10   3  10   5  10
11|  1  11  11  11  11  11  11  11  11  11  11   1  11  11
12|  1  12   6   4   9  12   4  12  12   8   6  12   6  12
13|  1  13  13  13  13  13  13  13  13  13  13  13  13   1
14|  1  14   7  14   7  14  14   2   7  14  14  14  14  14
15|  1  15  15   5  15   3  10  15  15  10  15  15   5  15
16|  1  16   8  16   4  16   8  16  10  16   8  16  12  16
In row n=5 the coefficients represent A(x) = 1 +5*x +5*x^2 +5*x^3 +5*x^4 +x^5 +5*x^6+... where A(x)^(1/5) = 1 +x -x^2 +3*x^3 -8*x^4 +24*x^5 -76*x^6... has only integer coefficients of A084205.

Maple

A109626aux := proc(n, k)
    option remember;
    local a, x, r ;
    if k =0 then
        1;
    else
        add( procname(n, j)*x^j, j=0..k-1) ;
        r := coeff(%^n, x, k) ;
        -floor((r-1)/n) ;
    end if;
end proc:
A109626 := proc(n, k)
    option remember;
    local j, x ;
    if k =0 or n = 1 then
        1;
    else
        add( A109626aux(n, j)*x^j, j=0..k) ;
        coeftayl(%^n, x=0, k) ;
    end if;
end proc:
seq(seq(A109626(d-k, k), k=0..d-1), d=1..12) ; # R. J. Mathar, May 25 2026

Mathematica

f[n_]:= f[n]= Block[{a}, a[0] = 1; a[l_]:= a[l]= Block[{k = 1, s = Sum[ a[i]*x^i, {i, 0, l-1}]}, While[ IntegerQ[Last[CoefficientList[Series[(s + k*x^l)^(1/n), {x, 0, l}], x]]] != True, k++ ]; k]; Table[a[j], {j, 0, 32}]];
T[n_, m_]:= f[n][[m]];
Flatten[Table[T[i, n-i], {n, 15}, {i, n-1, 1, -1}]]

Prog

(PARI) A109626_row(n, len=40)={my(A=1, m); vector(len, k, if(k>m=1, while(denominator(polcoeff(sqrtn(O(x^k)+A+=x^(k-1), n), k-1))>1, m++); m, 1))} \\ M. F. Hasler, Jan 27 2025

Crossrefs

Rows: A000012, A083952, A083953, A083954, A083945, A083946, A083947, A083948, A083949, A083950, A084066, A084067.

Columns: A000012, A111627, A026741, A051176, A111607, A060791, A111608, A106608, A111609, A111610, A111611, A106612, A106614, A106618, A106620.

Diagonals: A000027 (main), A111614 (first upper), A111627 (2nd), A111615 (3rd), A111618 (first lower), A111623 (2nd).

Other diagonals: A005408 (T(2*n-1, n)), A111626, A111627, A111620, A111629, A111630.

Cf. A111603, A111604, A084204 (n=4).

Sequence in context: A108888 A124021 A387657 * A182285 A160182 A195825

Adjacent sequences: A109623 A109624 A109625 * A109627 A109628 A109629

Keyword

nonn,tabl

Author

Paul D. Hanna and Robert G. Wilson v, Aug 01 2005

Status

approved