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%I #9 Jun 19 2016 07:38:53
%S 1,0,1,6,49,520,6841,107744,1979713,41611392,985263601,25958682112,
%T 753424361713,23888905963520,821659980883561,30472793606184960,
%U 1212264580564478209,51496393511442350080,2326573297949232710881,111398795962351731212288,5635038492335356268228401,300285949343202022103973888,16814498551154751682934232601,987042812055984079330393194496
%N a(n) = ((n+1)^(n-1) - (n-1)^(n-1))/2 for n>=1.
%F E.g.f.: cosh(x*W(x)) = (W(x) + 1/W(x))/2 where W(x) = LambertW(-x)/(-x) = exp(x*W(x)) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!.
%F a(n) = Sum_{k=0..floor((n-1)/2)} C(n-1,2*k+1) * n^(n-2*k-2).
%e E.g.f.: A(x) = 1 + x^2/2! + 6*x^3/3! + 49*x^4/4! + 520*x^5/5! + 6841*x^6/6! + 107744*x^7/7! + 1979713*x^8/8! + 41611392*x^9/9! + 985263601*x^10/10! +...
%e such that A(x) = cosh(x*W(x))
%e where W(x) = LambertW(-x)/(-x) begins
%e W(x) = 1 + x + 3*x^2/2! + 16*x^3/3! + 125*x^4/4! + 1296*x^5/5! + 16807*x^6/6! + 262144*x^7/7! + 4782969*x^8/8! + 100000000*x^9/9! +...+ (n+1)^(n-1)*x^n/n! +...
%e and satisfies W(x) = exp(x*W(x)).
%e Also, A(x) = (W(x) + 1/W(x))/2 where
%e 1/W(x) = 1 - x - x^2/2! - 4*x^3/3! - 27*x^4/4! - 256*x^5/5! - 3125*x^6/6! - 46656*x^7/7! - 823543*x^8/8! +...+ -(n-1)^(n-1)*x^n/n! +...
%o (PARI) {a(n) = ((n+1)^(n-1) - (n-1)^(n-1))/2}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = sum(k=0,(n-1)\2, binomial(n-1,2*k+1) * n^(n-2*k-2))}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(W=sum(m=0,n, (m+1)^(m-1)*x^m/m!) +x*O(x^n)); n!*polcoeff(cosh(x*W),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000272, A195136, A274279.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Jun 19 2016
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