A381195 Expansion of g.f. (1 - sqrt(1 - 1728*x))/(864*x).

1, 432, 373248, 403107840, 487599243264, 631928619270144, 857978513934778368, 1204601833564428828672, 1734626640332777513287680, 2547819609320783611516944384, 3802273336964543978787469000704, 5749037285490390495926653129064448, 8788066841328079995004188536982208512

Offset

0,2

Links

Table of n, a(n) for n=0..12.

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. 23.

Formula

a(n) = (-27)^n*2^(1+6*n)*binomial(1/2,1+n).

E.g.f.: exp(864*x)*(BesselI(0, 864*x) - BesselI(1, 864*x)).

D-finite with recurrence (n+1)*a(n) + 864*(-2*n+1)*a(n-1) = 0. - R. J. Mathar, Feb 18 2025

a(n) ~ 1728^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, May 29 2025

From Amiram Eldar, Nov 06 2025: (Start)

Sum_{n>=0} 1/a(n) = 2986848/2982529 + 4478976*arccosec(24*sqrt(3))/(2982529*sqrt(1727)).

Sum_{n>=0} (-1)^n/a(n) = 2985120/2989441 - 4478976*arccosech(24*sqrt(3))/(2989441*sqrt(1729)). (End)

Mathematica

CoefficientList[Series[(1-Sqrt[1-1728x])/(864x), {x, 0, 12}], x]

Crossrefs

Cf. A009971, A277757.

Sequence in context: A300053 A047804 A008691 * A269881 A193141 A360652

Adjacent sequences: A381192 A381193 A381194 * A381196 A381197 A381198

Keyword

nonn,easy

Author

Stefano Spezia, Feb 16 2025

Status

approved