A385304 Expansion of e.g.f. 1/(1 - 2 * sinh(x))^(1/2).

1, 1, 3, 16, 117, 1096, 12543, 169576, 2644617, 46735936, 922993083, 20145579136, 481555537917, 12511452674176, 351058439096823, 10579734482269696, 340820224678288017, 11687491783287586816, 425075150516293691763, 16343274366458168160256, 662325275389743380902917

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Peter Bala, The method of Graves for computing inverse functions.

Formula

a(n) = Sum_{k=0..n} A001147(k) * A136630(n,k).

a(n) ~ sqrt(2) * n^n / (5^(1/4) * exp(n) * log((1 + sqrt(5))/2)^(n + 1/2)). - Vaclav Kotesovec, Jun 28 2025

Maple

# makes use of Graves' method for computing inverse functions
d := proc (n, x) option remember; if n = 0 then 1/sqrt(1 - 2*x) else simplify( sqrt(1 + x^2)*diff(d(n-1, x), x)) end if end proc:
seq( eval( d(n, x), x = 0), n = 0..20); # Peter Bala, Feb 20 2026

Prog

(PARI) a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a001147(n) = prod(k=0, n-1, 2*k+1);
a(n) = sum(k=0, n, a001147(k)*a136630(n, k));

Crossrefs

Cf. A006154, A012103, A385305.

Cf. A380015, A385306, A385308, A385310.

Cf. A001147, A136630, A235134, A364822.

Sequence in context: A111555 A336293 A355718 * A385376 A221409 A074522

Adjacent sequences: A385301 A385302 A385303 * A385305 A385306 A385307

Keyword

nonn,easy

Author

Seiichi Manyama, Jun 24 2025

Status

approved