A201627 - OEIS

%I #11 Jul 07 2012 19:08:04

%S 1,1,4,29,312,4481,80768,1754549,44647040,1303097665,42923116032,

%T 1575332861101,63754405679104,2820829737123841,135469202252333056,

%U 7018336152909163205,390175030207597805568,23169468447962190613121,1463683656780476860989440,98016257612539018485477821

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

%C Coefficients in the expansion of 1/(1-sin(x)) yield the Euler numbers (A000111).

%F E.g.f. A(x) satisfies: A( x*(1 - sin(x)) ) = 1/(1 - sin(x)).

%F E.g.f.: (1/x)*Series_Reversion( x*(1 - sin(x)) ).

%F a(n) = [x^n] 1/(1 - sin(x))^(n+1) / (n+1).

%F a(n) = A214222(n+1)/(n+1).

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 29*x^3/3! + 312*x^4/4! + 4481*x^5/5! +...

%e The coefficients in initial powers of G(x) = 1/(1 - sin(x)) begin:

%e G^1: [(1), 1, 2, 5, 16, 61, 272, 1385, 7936, ..., A000111(n+1), ...];

%e G^2: [1,(2), 6, 22, 96, 482, 2736, 17302, ...];

%e G^3: [1, 3,(12), 57, 312, 1923, 13152, 98697, ...];

%e G^4: [1, 4, 20,(116), 760, 5524, 44000, 380516, ...];

%e G^5: [1, 5, 30, 205,(1560), 13025, 118080, 1153105, ...];

%e G^6: [1, 6, 42, 330, 2856,(26886), 272832, 2963850, ...];

%e G^7: [1, 7, 56, 497, 4816, 50407, (565376), 6754097, ...];

%e G^8: [1, 8, 72, 712, 7632, 87848, 1078272,(14036392), ...]; ...

%e where coefficients in parenthesis form initial terms of this sequence:

%e [1/1, 2/2, 12/3, 116/4, 1560/5, 26886/6, 565376/7, 14036392/8, ...].

%o (PARI) {a(n)=n!*polcoeff(1/x*serreverse(x*(1-sin(x+x^2*O(x^n)))),n)}

%o (PARI) {a(n)=n!*polcoeff(1/(1-sin(x+x*O(x^n)))^(n+1)/(n+1), n)}

%Y Cf. A214222, A201594, A000111.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Dec 03 2011