A201628 E.g.f. satisfies: A(x) = 1/(1 - sinh(x*A(x))).

1, 1, 4, 31, 360, 5601, 109568, 2586151, 71555200, 2271961825, 81441188352, 3253620672303, 143361363439616, 6907049546879041, 361245668908466176, 20383791705206338807, 1234336634416972726272, 79843983527411321710401, 5494767253686351671459840, 400863405346004202504321343

Offset: 0

Author

Paul D. Hanna, Dec 03 2011

Keywords

• nonn

Comments

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.

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 31*x^3/3! + 360*x^4/4! + 5601*x^5/5! +...
The coefficients in initial powers of G(x) = 1/(1 - sinh(x)) begin:
G^1: [(1), 1, 2, 7, 32, 181, 1232, 9787, 88832, ..., A006154(n), ...];
G^2: [1,(2), 6, 26, 144, 962, 7536, 67706, ...];
G^3: [1, 3,(12), 63, 408, 3123, 27552, 275103, ...];
G^4: [1, 4, 20,(124), 920, 7924, 77600, 850924, ...];
G^5: [1, 5, 30, 215,(1800), 17225, 185280, 2211515, ...];
G^6: [1, 6, 42, 342, 3192,(33606), 393792, 5080662, ...];
G^7: [1, 7, 56, 511, 5264, 60487, (766976), 10634911, ...];
G^8: [1, 8, 72, 728, 8208, 102248, 1395072,(20689208), ...]; ...
where coefficients in parenthesis form initial terms of this sequence:
[1/1, 2/2, 12/3, 124/4, 1800/5, 33606/6, 766976/7, 20689208/8, ...].
		

Crossrefs

Cf. A214223, A201627, A201595, A006154.

Programs

PARI

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

PARI

{a(n)=n!*polcoeff(1/(1 - sinh(x+x*O(x^n)))^(n+1)/(n+1), n)}

Formula

E.g.f. A(x) satisfies: A( x*(1 - sinh(x)) ) = 1/(1 - sinh(x)).

E.g.f.: (1/x)*Series_Reversion( x*(1 - sinh(x)) ).

a(n) = [x^n] 1/(1 - sinh(x))^(n+1) / (n+1).

a(n) = A214223(n+1)/(n+1).