OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..49
FORMULA
E.g.f. A(x) = C(x,y=x) given C(x,y) = 1 + Integral S(x,y)*C(y,x) dx such that C(x,y)^2 - S(x,y)^2 = 1 and C(y,x) = Integral S(y,x)*C(x,y) dy, where A(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)!.
EXAMPLE
E.g.f.: A(x) = 1 + x^2/2! + 13*x^4/4! + 301*x^6/6! + 13049*x^8/8! + 916441*x^10/10! + 94195333*x^12/12! + 13347584069*x^14/14! + 2494336502897*x^16/16! + 594306468307633*x^18/18! + ...
RELATED SERIES.
sqrt(A(x)^2 - 1) = x + 4*x^3/3! + 56*x^5/5! + 1856*x^7/7! + 103936*x^9/9! + 8893952*x^11/11! + 1080485888*x^13/13! + 176673603584*x^15/15! + 37417114009600*x^17/17! + 9963927777050624*x^19/19! + ... + A322733(n)*x^(2*n+1)/(2*n+1)! + ...
A(x) = cosh( Integral D(x) dx ) where D(x) = A'(x)/sqrt(A(x)^2 - 1) begins
D(x) = 1 + 3*x^2/2! + 25*x^4/4! + 595*x^6/6! + 26193*x^8/8! + 1832611*x^10/10! + 188365801*x^12/12! + 26696014003*x^14/14! + 4988672502305*x^16/16! + 1188611267890243*x^18/18! + ...
PROG
(PARI) {A322731(n, k) = my(Sx=x, Sy=y, Cx=1, Cy=1); for(i=1, 2*n,
Sx = intformal( Cx*Cy +x*O(x^(2*n)), x);
Cx = 1 + intformal( Sx*Cy, x);
Sy = intformal( Cy*Cx +y*O(y^(2*k)), y);
Cy = 1 + intformal( Sy*Cx, y));
(2*n)! *polcoeff(polcoeff(Cx, 2*n-2*k, x), 2*k, y)}
a(n) = sum(k=0, n, A322731(n, k))
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 01 2019
STATUS
approved