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%I #8 Sep 04 2022 08:09:49
%S 0,1,-2,9,-64,620,-7536,109032,-1809984,33562944,-681799680,
%T 14980204800,-354016189440,9017296704000,-249422713344000,
%U 7530733353024000,-246212297533440000,8509848430274150400,-302719894872204902400
%N E.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2) with A(0)=0.
%H G. C. Greubel, <a href="/A179199/b179199.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f. A=A(x) satisfies: x = A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! + ... where Dx(F) = d/dx(x*F).
%F ...
%F a(n) = -n*(n-2)!*Sum_{i=1..n-1} C(n-i+1,i+1)*a(n-i)/(n-i)! for n>1 with a(1)=1.
%F ...
%F a(n) = (-1)^(n-1)*n*A005119(n), where A005119 describes the infinitesimal generator of (x+x^2).
%F ...
%F Equals column 0 of A179198, the matrix log of triangle A030528, where A030528(n,k) = C(k,n-k); the g.f. of column k in A030528 is (x+x^2)^(k+1)/x.
%F ...
%F A179198(n,k) = (k+1)*a(n-k)/(n-1)! for n>0, k>=0, where A179198 = matrix log of triangle A030528.
%F ...
%e E.g.f.: A(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 620*x^5/5! - 7536*x^6/6! + 109032*x^7/7! - 1809984*x^8/8! + 33562944*x^9/9! - 681799680*x^10/10! + 14980204800*x^11/11! - 354016189440*x^12/12! + ...
%e E.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2) where:
%e . A(x+x^2) = x - 3*x^3/3! + 20*x^4/4! - 120*x^5/5! + 624*x^6/6! - 840*x^7/7! - 58752*x^8/8! + 1512000*x^9/9! - 25660800*x^10/10! + ...
%e E.g.f. A = A(x) satisfies:
%e . x = A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! + ...
%e where Dx(F) = d/dx(x*F) and expansions begin:
%e . A*Dx(A) = 4*x^2/2! - 30*x^3/3! + 288*x^4/4! - 3500*x^5/5! +- ...
%e . A*Dx(A*Dx(A)) = 36*x^3/3! - 624*x^4/4! + 10680*x^5/5! -+ ...
%e . A*Dx(A*Dx(A*Dx(A))) = 576*x^4/4! - 18480*x^5/5! + 504000*x^6/6! -+ ...
%e . A*Dx(A*Dx(A*Dx(A*Dx(A)))) = 14400*x^5/5! - 751680*x^6/6! +- ...
%t a[n_]:= a[n]= If[n<2, n, (-1/(n-1))*Sum[j!*Binomial[n, j]*Binomial[n-j+1, j+1]*a[n -j], {j, n-1}]];
%t Table[a[n], {n,0,40}] (* _G. C. Greubel_, Sep 03 2022 *)
%o (PARI) /* E.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2): */
%o {a(n)=local(A=x,B);for(m=2,n,B=(1+x)/(1+2*x+O(x^(n+3)))*subst(A,x,x+x^2+O(x^(n+3)));A=A-polcoeff(B,m+1)*x^m/(m-1));n!*polcoeff(A,n)}
%o (PARI) /* Recurrence (slow): */
%o {a(n)=if(n<1, 0, if(n==1, 1, -n*(n-2)!*sum(i=1, n-1,binomial(n-i+1, i+1)*a(n-i)/(n-i)!)))}
%o (PARI) /* x = A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...: */
%o {a(n)=local(A=x+sum(m=2,n-1,a(m)*x^m/m!),G=1,R=0);R=sum(m=1,n,(G=A*deriv(x*G+x*O(x^n)))/m!);if(n==1,1,-n!*polcoeff(R,n))}
%o (PARI) /* As column 0 of the matrix log of triangle A030528: */
%o {a(n)=local(A030528=matrix(n+1,n+1,r,c,if(r>=c,binomial(c,r-c))),LOG,ID=A030528^0);LOG=sum(m=1,n+1,-(ID-A030528)^m/m);n!*LOG[n+1,1]}
%o (SageMath)
%o @CachedFunction
%o def a(n): # a = A179199
%o if (n<2): return n
%o else: return (-1/(n-1))*sum( factorial(j)*binomial(n,j)*binomial(n-j+1, j+1)*a(n-j) for j in (1..n-1) )
%o [a(n) for n in (0..40)] # _G. C. Greubel_, Sep 03 2022
%Y Cf. A005119, A030528, A179198.
%K eigen,sign
%O 0,3
%A _Paul D. Hanna_, Jul 09 2010
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