A281436 E.g.f. C(x) satisfies: C(x) = cosh( Integral C(x)^6 dx ).

1, 1, 25, 1921, 301105, 79715041, 31953352585, 18055521444961, 13675020394100065, 13371875150000047681, 16399205085221871057145, 24649278939856374854251201, 44562940283889239727355265425, 95400484808911059939339873215521, 238677467614161144737008148697894505, 690019785744332320572996561216607892641, 2282793490173453501140213827073151244641985

Offset

0,3

Links

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

Formula

C(x)^2 - S(x)^2 = 1 and C(x) = 1 + Integral C(x)^6*S(x) dx, where S(x) is described by A281435.

From Peter Bala, Dec 23 2025: (Start)

The g.f. C(x) is algebraic: (225*x^2 - 64)*C(x)^10 + 40*C(x)^4 + 15*C(x)^2 + 9 = 0 with C(0) = 1.

For n >= 1, define f(n, x) = sec^5(x)*d/dx(f(n-1, x)) with f(0, x) = sec(x). Then a(n) = f(2*n, 0)? (End)

Example

C(x) = 1 + x^2/2! + 25*x^4/4! + 1921*x^6/6! + 301105*x^8/8! + 79715041*x^10/10! + 31953352585*x^12/12! + 18055521444961*x^14/14! + 13675020394100065*x^16/16! + 13371875150000047681*x^18/18! + .... - Peter Bala, Dec 23 2025

Prog

(PARI) {a(n) = my(S=x, C=1); for(i=0, n, S = intformal( C^7 +x*O(x^(2*n))); C = 1 + intformal( S*C^6 ) ); (2*n)!*polcoeff(C, 2*n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A281180, A281181, A281183, A281184, A281427, A281428, A281432.

Sequence in context: A322247 A177837 A056047 * A197671 A051112 A061843

Adjacent sequences: A281433 A281434 A281435 * A281437 A281438 A281439

Keyword

nonn,easy

Author

Paul D. Hanna, Jan 21 2017

Status

approved