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G.f.: x = Sum_{n>=1} a(n)*x^n * Sum_{k=0..n} (k+1)*binomial(n,k)*(-x)^k.
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%I #10 Jul 05 2014 07:11:30

%S 1,2,8,42,264,1916,15744,144546,1467544,16335972,197916768,2593286692,

%T 36547123728,551308006392,8863973173120,151328667184530,

%U 2734113089546040,52120618871199060,1045503182929422240,22014065919151444140,485475449631284066160

%N G.f.: x = Sum_{n>=1} a(n)*x^n * Sum_{k=0..n} (k+1)*binomial(n,k)*(-x)^k.

%H Vaclav Kotesovec, <a href="/A182520/b182520.txt">Table of n, a(n) for n = 1..300</a>

%F G.f.: C(x) + Sum_{n>=2} (n+1)!/3! * C(x)^n, where C(x) = (1-sqrt(1-4*x))/2 is a g.f. of the Catalan numbers A000108.

%F Recurrence: (n-1)*n*a(n) = (n-1)*(n^2 + 9*n - 19)*a(n-1) - 2*(n-2)*(4*n^2 + 2*n - 31)*a(n-2) + 4*(n-2)*(2*n-7)*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Jul 05 2014

%F a(n) ~ exp(1)/6 * n * n!. - _Vaclav Kotesovec_, Jul 05 2014

%e G.f.: A(x) = x + 2*x^2 + 8*x^3 + 42*x^4 + 264*x^5 + 1916*x^6 + 15744*x^7 +...

%e such that

%e A(x) = C(x) + C(x)^2 + 4*C(x)^3 + 20*C(x)^4 + 120*C(x)^5 +...+ (n+1)!/3!*C(x)^n +...

%e where

%e C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 +...+ A000108(n)*x^(n+1) +...

%e By definition, the terms a(n) satisfy:

%e x = a(1)*x*(1-2*x) + a(2)*x^2*(1-4*x+3*x^2) + a(3)*x^3*(1-6*x+9*x^2-4*x^3) + a(4)*x^4*(1-8*x+18*x^2-16*x^3+5*x^4) +...

%o (PARI) {a(n)=if(n<1, 0, polcoeff(x-sum(m=1, n-1, a(m)*x^m*sum(k=0, m, (k+1)*binomial(m, k)*(-x)^k)+x*O(x^n)), n))}

%o (PARI) {a(n)=local(C=serreverse(x-x^2+x^2*O(x^n)));polcoeff(C+sum(m=2,n,(m+1)!/3!*C^m),n)}

%o for(n=0,30,print1(a(n),", "))

%Y Cf. A000108.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 03 2012