A381272 - OEIS

A381272

a(n) = denominator( [x*y*z]^n 1/sqrt((1 - (x + y + z))*(1 - x - y - z^2)) ).

1, 1, 4, 1, 64, 64, 64, 32, 16384, 16384, 65536, 4096, 262144, 262144, 524288, 262144, 1073741824, 1073741824, 4294967296, 1073741824, 68719476736, 68719476736, 17179869184, 8589934592, 4398046511104, 4398046511104, 17592186044416, 2199023255552, 140737488355328, 140737488355328, 281474976710656, 140737488355328, 4611686018427387904

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OFFSET

0,3

COMMENTS

a(n) = 2^m for some m <= 2*n - 2, where the upper bound is attained when n is a power of 2. - Max Alekseyev, Apr 01 2025

LINKS

Table of n, a(n) for n=0..32.

S. Hassani, J.-M. Maillard, and N. Zenine, On the diagonals of rational functions: the minimal number of variables (unabridged version), arXiv:2502.05543 [math-ph], 2025. See p. 28.

FORMULA

[x*y*z]^n 1/sqrt((1 - (x + y + z))*(1 - x - y - z^2)) = C(2*n,n) * Sum_{k=0..n} C(-1/2,k) * Sum_{j=0..k} C(k,j) * C(k-j,n-k-2*j) * C(-(k+j+1),2*n) * (-1)^(k-j). - Max Alekseyev, Apr 01 2025

MATHEMATICA

a[n_]:=Denominator[SeriesCoefficient[1/Sqrt[(1-(x+y+z))(1-x-y-z^2)], {x, 0, n}, {y, 0, n}, {z, 0, n}]]; Array[a, 10, 0]

PROG

(PARI) a381272(n) = denominator( binomial(2*n, n) * sum(k=0, n, binomial(-1/2, k) * sum(j=0, k, binomial(k, j) * binomial(k-j, n-k-2*j) * binomial(-(k+j+1), 2*n) * (-1)^(k-j) )) ); \\ Max Alekseyev, Apr 01 2025

CROSSREFS

Cf. A381271 (numerator).

Cf. A381269, A381270.

Sequence in context: A338681 A069740 A173008 * A298828 A114917 A100864

Adjacent sequences: A381269 A381270 A381271 * A381273 A381274 A381275

KEYWORD

nonn,frac

AUTHOR

Stefano Spezia, Feb 18 2025

EXTENSIONS

Terms a(14) onward from Max Alekseyev, Apr 01 2025

STATUS

approved