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%I #42 Apr 10 2019 04:53:03
%S 1,1,4,28,288,3936,67328,1385728,33372160,921118720,28677169152,
%T 994360565760,38007586684928,1587878686621696,71990467473965056,
%U 3520403893852831744,184707311409882464256,10350444842488122310656,616975843658373414256640,38981881007475178476666880
%N E.g.f. satisfies: A(x) = exp(x*A(x)) * cosh(x*A(x)).
%H G. C. Greubel, <a href="/A201595/b201595.txt">Table of n, a(n) for n = 0..370</a>
%H Nantel Bergeron, Laura Colmenarejo, Shu Xiao Li, John Machacek, Robin Sulzgruber, Mike Zabrocki, Adriano Garsia, Marino Romero, Don Qui, Nolan Wallach, <a href="http://garsia.math.yorku.ca/~zabrocki/summary.pdf">Super Harmonics and a representation theoretic model for the Delta conjecture</a>, A summary of the open problem sessions of Jan 24, 2019, Representation Theory Connections to (q,t)-Combinatorics (19w5131), Banff, BC, Canada.
%F a(n) = (1/2) * Sum_{k=0..n+1} C(n+1,k) * k^n / (n+1).
%F a(n) = [x^n/n!] exp((n+1)*x) * cosh(x)^(n+1) / (n+1).
%F E.g.f. A(x) satisfies:
%F (1) A( x*exp(-x)/cosh(x) ) = exp(x)*cosh(x).
%F (2) A(x) = (1/x)*Series_Reversion( x*exp(-x)/cosh(x) ).
%F (3) A(x) = (1 + exp(2*x*A(x)))/2.
%F (4) A(x) = exp(G(x)) where G(x) is the e.g.f. of A074932.
%F (5) A(x) = Sum_{n>=0} (n+1)^(n-1) * cosh((n+1)*x) * x^n/n!. - _Paul D. Hanna_, Oct 24 2012
%F (6) A(x) = 1 + Sum_{n>=1} n^n * sinh(n*x)/(n*x) * x^n/n!. - _Paul D. Hanna_, Nov 20 2012
%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then
%F a(n,m) = Sum_{k=0..n+m} C(n+m, k) * k^n * m/(n+m) / 2^m.
%F a(n) = A214225(n+1)/(n+1).
%F E.g.f.: (x-LambertW(-x*exp(x)))/(2*x). - _Vaclav Kotesovec_, Dec 04 2012
%F a(n) ~ n!*sqrt(LambertW(exp(-1))+1)/(2*sqrt(2*Pi)*n^(3/2)*LambertW(exp(-1))^(n+1)). - _Vaclav Kotesovec_, Dec 04 2012
%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 28*x^3/3! + 288*x^4/4! + 3936*x^5/5! +...
%e The coefficients of x^n/n! in initial powers of G(x) = (1 + exp(2*x))/2 begin:
%e G^1: [(1), 1, 2, 4, 8, 16, 32, 64, 128, ...];
%e G^2: [1,(2), 6, 20, 72, 272, 1056, 4160, ...];
%e G^3: [1, 3,(12), 54, 264, 1368, 7392, 41184, ...];
%e G^4: [1, 4, 20,(112), 680, 4384, 29600, 207232, ...];
%e G^5: [1, 5, 30, 200,(1440), 11000, 88080, 732800, ...];
%e G^6: [1, 6, 42, 324, 2688,(23616), 217392, 2080224, ...];
%e G^7: [1, 7, 56, 490, 4592, 45472,(471296), 5076400, ...];
%e G^8: [1, 8, 72, 704, 7344, 80768, 928512,(11085824), ...]; ...
%e where coefficients in parenthesis form initial terms of this sequence:
%e [1/1, 2/2, 12/3, 112/4, 1440/5, 23616/6, 471296/7, 11085824/8, ...].
%t Join[{1},Table[Sum[Binomial[n+1,k] k^n/(n+1),{k,0,n+1}]/2,{n,20}]] (* _Harvey P. Dale_, Feb 04 2012 *)
%t CoefficientList[Series[(x-LambertW[-x*E^x])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Dec 04 2012 *)
%o (PARI) a(n)=n!*polcoeff(1/x*serreverse(x*exp(-x+x^2*O(x^n))/cosh(x+x^2*O(x^n))),n)
%o (PARI) a(n)=local(X=x+x*O(x^n));n!*polcoeff(exp((n+1)*X)*cosh(X)^(n+1)/(n+1),n)
%o (PARI) a(n)=sum(k=0,n+1,binomial(n+1,k)*k^n/(n+1)/2)
%o (PARI) /* Formula for a(n,m) where A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!: */
%o {a(n,m=1)=sum(k=0, n+m, binomial(n+m, k)*k^n*m/(n+m)/2^m)}
%o (PARI) /* Formula derived from a LambertW identity: */
%o {a(n)=local(A=sum(k=0,n,(k+1)^(k-1)*cosh((k+1)*x+x*O(x^n))*x^k/k!));n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Oct 24 2012
%o (PARI) /* Formula derived from a LambertW identity: */
%o {a(n)=local(A=1+sum(k=1,n,k^k*sinh(k*x+x^2*O(x^n))/(k*x)*x^k/k!));n!*polcoeff(A,n)}
%o for(n=0,20,print1(a(n),", ")) \\ _Paul D. Hanna_, Nov 20 2012
%Y Cf. A214225, A201594, A074932.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Dec 03 2011
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