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%I #9 Jul 07 2012 19:03:55
%S 1,1,4,31,360,5601,109568,2586151,71555200,2271961825,81441188352,
%T 3253620672303,143361363439616,6907049546879041,361245668908466176,
%U 20383791705206338807,1234336634416972726272,79843983527411321710401,5494767253686351671459840,400863405346004202504321343
%N E.g.f. satisfies: A(x) = 1/(1 - sinh(x*A(x))).
%C The function 1/(1-sinh(x)) is the e.g.f. of A006154, where A006154(n) is the number of labeled ordered partitions of an n-set into odd parts.
%F E.g.f. A(x) satisfies: A( x*(1 - sinh(x)) ) = 1/(1 - sinh(x)).
%F E.g.f.: (1/x)*Series_Reversion( x*(1 - sinh(x)) ).
%F a(n) = [x^n] 1/(1 - sinh(x))^(n+1) / (n+1).
%F a(n) = A214223(n+1)/(n+1).
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! +...
%e The coefficients in initial powers of G(x) = 1/(1 - sinh(x)) begin:
%e G^1: [(1), 1, 2, 7, 32, 181, 1232, 9787, 88832, ..., A006154(n), ...];
%e G^2: [1,(2), 6, 26, 144, 962, 7536, 67706, ...];
%e G^3: [1, 3,(12), 63, 408, 3123, 27552, 275103, ...];
%e G^4: [1, 4, 20,(124), 920, 7924, 77600, 850924, ...];
%e G^5: [1, 5, 30, 215,(1800), 17225, 185280, 2211515, ...];
%e G^6: [1, 6, 42, 342, 3192,(33606), 393792, 5080662, ...];
%e G^7: [1, 7, 56, 511, 5264, 60487, (766976), 10634911, ...];
%e G^8: [1, 8, 72, 728, 8208, 102248, 1395072,(20689208), ...]; ...
%e where coefficients in parenthesis form initial terms of this sequence:
%e [1/1, 2/2, 12/3, 124/4, 1800/5, 33606/6, 766976/7, 20689208/8, ...].
%o (PARI) {a(n)=n!*polcoeff(1/x*serreverse(x*(1-sinh(x+x^2*O(x^n)))),n)}
%o (PARI) {a(n)=n!*polcoeff(1/(1 - sinh(x+x*O(x^n)))^(n+1)/(n+1), n)}
%Y Cf. A214223, A201627, A201595, A006154.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 03 2011
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