A290579 - OEIS

%I #13 Aug 20 2017 06:32:11

%S 1,1,3,17,145,1661,23947,416765,8504865,199200985,5268047411,

%T 155282897065,5048546868209,179483438440021,6927036355244475,

%U 288439692143197397,12889578930617144641,615304922810783744561,31249797537098034966499,1682478707717022082757441,95721167135792642044679505,5738213812249917154871122861,361520344958259229318337826923,23881324878748364254507881173677

%N E.g.f. satisfies: A(x) = (1 + sin(x*A(x))) / cos(x*A(x)).

%C Equals row sums of irregular triangle A290580.

%C A diagonal in square array A185414.

%H Vaclav Kotesovec, <a href="/A290579/b290579.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f.: A(x) = (1/x) * Series_Reversion( x*cos(x)/(1 + sin(x)) ).

%F E.g.f. satisfies: A(x) = Sum_{n>=0} A000111(n) * x^n * A(x)^n / n!.

%F E.g.f.: A(x) = exp( sum_{n>=1} A185414(n,n)*x^n/n! ).

%F a(n) = A185414(n,n+1) for n>=1.

%F a(n) = [x^n/n!] ( (1 + sin(x)) / cos(x) )^(n+1) / (n+1) for n>=0.

%F a(n) ~ n^(n-1) / (exp(n) * r^(n+1/2)), where r = 0.3263879708167851846572540343... is the root of the equation r = 1 - sin(sqrt(r*(2-r))). - _Vaclav Kotesovec_, Aug 20 2017

%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 145*x^4/4! + 1661*x^5/5! + 23947*x^6/6! + 416765*x^7/7! + 8504865*x^8/8! +...

%e such that A(x) = (1 + sin(x*A(x))) / cos(x*A(x)).

%e RELATED SERIES.

%e (1 + sin(x)) / cos(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 16*x^5/5! + 61*x^6/6! + 272*x^7/7! + 1385*x^8/8! +...+ A000111(n)*x^n/n! +...

%e where A000111 is the sequence of zigzag numbers.

%e Given e.g.f. A(x), then the logarithm of A(x) begins:

%e log(A(x)) = x + 2*x^2/2! + 10*x^3/3! + 80*x^4/4! + 880*x^5/5! + 12336*x^6/6! + 210320*x^7/7! + 4225024*x^8/8! +...+ A185414(n,n)*x^n/n! +...

%o (PARI) {a(n) = my(A=1); for(i=0,n, A = (1 + sin(x*A)) / cos(x*A +x*O(x^n)) ); n!*polcoeff(A,n)}

%o for(n=0,25, print1(a(n),", "))

%o (PARI) {a(n) = my(X=x +O(x^(n+2))); A = (1/x) * serreverse( x*cos(X)/(1 + sin(X)) ); n!*polcoeff(A,n)}

%o for(n=0,25, print1(a(n),", "))

%Y Cf. A000111, A290580, A185414.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Aug 06 2017