A326328 - OEIS

A326328

a(n) = (2*n)! [x^(2*n)] cosh(x)^(-3).

1, -3, 33, -723, 25953, -1376643, 101031873, -9795436563, 1212135593793, -186388033956483, 34859622790687713, -7791941518975112403, 2051293521728340489633, -628173356956461494680323, 221398076445213367209575553, -88980467736394156270609236243, 40450409313733718675802456121473

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OFFSET

0,2

COMMENTS

Apparently all terms except the initial 1 have 3-valuation 1. - F. Chapoton, Nov 25 2021

REFERENCES

H. S. Wall, Analytic Theory of Continued Fractions, Chelsea 1973, p. 206.

LINKS

Table of n, a(n) for n=0..16.

FORMULA

O.g.f. as a Stieltjes-type continued fraction: 1/(1 + 3*x/(1 + 8*x/(1 + 15*x/(1 +... + n*(n + 2)*x/(1 + ... ))))). See Wall, Chapter XI, eqn. 53.11 with k = 3. - Peter Bala, Dec 13 2025

From Seiichi Manyama, Apr 21 2026: (Start)

a(0) = 1; a(n) = -(1/4) * Sum_{k=0..n-1} (3 + 9^(n-k)) * binomial(2*n,2*k) * a(k).

a(n) = ((-1)^n/2) * (A000364(n) + A000364(n+1)) = (1/2) * (A028296(n) - A028296(n+1)). (End)

MAPLE

egf := cosh(z)^(-3): ser := series(egf, z, 36):

seq((2*n)!*coeff(ser, z, 2*n), n=0..16);

PROG

(PARI) a(n) = my(x='x+O('x^(2*n+1))); (2*n)!*polcoef(1/cosh(x)^3, 2*n); \\ Seiichi Manyama, Apr 21 2026

CROSSREFS

Row 3 of A326327.

Cf. A000364, A028296.

Sequence in context: A380722 A091462 A340971 * A233319 A003715 A247030