A179199 E.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2) with A(0)=0.
0, 1, -2, 9, -64, 620, -7536, 109032, -1809984, 33562944, -681799680, 14980204800, -354016189440, 9017296704000, -249422713344000, 7530733353024000, -246212297533440000, 8509848430274150400, -302719894872204902400
Offset: 0
Examples
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.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2) where: . 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.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) and expansions begin: . A*Dx(A) = 4*x^2/2! - 30*x^3/3! + 288*x^4/4! - 3500*x^5/5! +- ... . A*Dx(A*Dx(A)) = 36*x^3/3! - 624*x^4/4! + 10680*x^5/5! -+ ... . A*Dx(A*Dx(A*Dx(A))) = 576*x^4/4! - 18480*x^5/5! + 504000*x^6/6! -+ ... . A*Dx(A*Dx(A*Dx(A*Dx(A)))) = 14400*x^5/5! - 751680*x^6/6! +- ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..300
- Loïc Foissy, Cointeraction on noncrossing partitions and related polynomial invariants, arXiv:2501.18212 [math.CO], 2025. See p. 24.
Programs
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Mathematica
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}]]; Table[a[n], {n,0,40}] (* G. C. Greubel, Sep 03 2022 *) -
PARI
/* E.g.f. satisfies: A(x) = (1+x)/(1+2*x)*A(x+x^2): */ {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)} -
PARI
/* Recurrence (slow): */ {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)!)))} -
PARI
/* x = A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! +...: */ {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))} -
PARI
/* As column 0 of the matrix log of triangle A030528: */ {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]}
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SageMath
@CachedFunction def a(n): # a = A179199 if (n<2): return n 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) ) [a(n) for n in (0..40)] # G. C. Greubel, Sep 03 2022
Formula
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).
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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.
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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.
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