|
%I #18 Sep 03 2022 08:49:59
%S 0,2,-2,6,-28,160,-936,4536,-20448,627264,-19699200,43908480,
%T 17788273920,-211715112960,-41219197125120,1301670191808000,
%U 160057006683033600,-10037518414724505600,-1007362871616478003200
%N E.g.f. satisfies: A(x) = A( x/(1-x)^2 ) * (1-x)/(1+x) with A(0)=0.
%H Paul D. Hanna, <a href="/A179320/b179320.txt">Table of n, a(n) for n = 0..200</a>
%H J. Cigler, <a href="https://homepage.univie.ac.at/Johann.Cigler/preprints/q-pol.pdf">Some results and conjectures about a class of q-polynomials with simple moments</a>, 2014.
%F E.g.f. A = A(x) satisfies:
%F (1) 1/(1-x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! + ...
%F (2) Catalan(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+ ...
%F (3) (1-x)^2/(1-3*x+x^2)^2 = 1 + 2*A + 2^2*A*Dx(A)/2! + 2^3*A*Dx(A*Dx(A))/3! + 2^4*A*Dx(A*Dx(A*Dx(A)))/4! + ...
%F where Dx(F) = d/dx(x*F).
%F INVERSION FORMULA:
%F More generally, if A(x) = A(G(x))*G(x)/(x*G'(x)) with G(0)=0, G'(0)=1,
%F then G(x) can be obtained from A=A(x) by the series:
%F G(x)/x = 1 + 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 ITERATION FORMULA:
%F Let G_{n}(x) denote the n-th iteration of G(x) = x/(1-x)^2, and A=A(x), then:
%F G_{n}(x)/x = 1 + n*A + n^2*A*Dx(A)/2! + n^3*A*Dx(A*Dx(A))/3! + n^4*A*Dx(A*Dx(A*Dx(A)))/4! + ...
%F MATRIX LOG OF RIORDAN ARRAY (G(x)/x, G(x)) where G(x) = x/(1-x)^2:
%F E.g.f. A(x) forms column 0 of A179321, the matrix log of triangle A078812, where A078812(n,k) = C(n+k+1,n-k); the g.f. of column k in A078812 is [x/(1-x)^2]^(k+1)/x.
%F A179321(n,k) = (k+1)*a(n-k)/(n-k)! for n>0, k>=0, where A179321 = matrix log of triangle A078812.
%F ...
%F a(n) = (-1)^(n-1)*2*A027614(n), where A027614 is related to Clebsch-Gordan formulas.
%F a(n) = 2*n!*b(n), with a(0) = 0, where b(n) = (-1/(2*(n-1))) * Sum_{j=2..2*floor(n/2)} A123521(n, j)*b(n-j+1), and b(1) = 1. - _G. C. Greubel_, Sep 01 2022
%e E.g.f.: A(x) = 2*x - 2*x^2/2! + 6*x^3/3! - 28*x^4/4! + 160*x^5/5! - 936*x^6/6! + 4536*x^7/7! - 20448*x^8/8! + 627264*x^9/9! - 19699200*x^10/10! + 43908480*x^11/11! + 17788273920*x^12/12! -+ ...
%e A(x/(1-x)^2) = 2*x + 6*x^2/2! + 18*x^3/3! + 68*x^4/4! + 360*x^5/5! + 2184*x^6/6! + 13272*x^7/7! + 122016*x^8/8! + 1541376*x^9/9! + 1987200*x^10/10! - 150923520*x^11/11! + 16504093440*x^12/12! + ...
%e where A(x/(1-x)^2) = (1+x)/(1-x)*A(x).
%e ...
%e Related expansions begin:
%e . A = 2*x - 2*x^2/2! + 6*x^3/3! - 28*x^4/4! + 160*x^5/5! + ...
%e . A*Dx(A)/2! = 8*x^2/2! - 30*x^3/3! + 180*x^4/4! - 1400*x^5/5! + ...
%e . A*Dx(A*Dx(A))/3! = 48*x^3/3! - 416*x^4/4! + 4280*x^5/5! + ...
%e . A*Dx(A*Dx(A*Dx(A)))/4! = 384*x^4/4! - 6160*x^5/5! + 98400*x^6/6! -+ ...
%e . A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 3840*x^5/5! - 100224*x^6/6! +- ...
%e where Catalan(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! +- ... = 1 - 2*x + 5*x^2 - 14*x^3 + 42*x^4 + ... + A000108(n)*(-x)^n + ...
%t T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, 2*(n-1), T[n-2, k-2] + Binomial[2n-k-1, 2n-2k-1] ]]; (* T = A123521 *)
%t b[n_]:= b[n]= If[n==1, 1, (-1/(2*(n-1)))*Sum[b[n-j+1]*T[n,j], {j, 2, 2*Floor[n/2]}]];
%t A179320[n_] := 2*n!*b[n];
%t Table[A179320[n], {n, 0, 40}] (* _G. C. Greubel_, Sep 01 2022 *)
%o (PARI) /* E.g.f. satisfies: A(x) = (1-x)/(1+x)*A(x/(1-x)^2): */
%o {a(n)=local(A=2*x, B); for(m=2, n, B=(1-x)/(1+x+O(x^(n+3)))*subst(A,x,x/(1-x+O(x^(n+3)))^2); A=A-polcoeff(B, m+1)*x^m/(m-1)/2); n!*polcoeff(A, n)}
%o (PARI) /* 1/(1-x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */
%o {a(n)=local(A=0+sum(m=1,n-1,a(m)*x^m/m!),D=1,R=0);R=-1/(1-x+x*O(x^n))^2+1+sum(m=1,n,(D=A*deriv(x*D+x*O(x^n)))/m!);-n!*polcoeff(R,n)}
%o (PARI) /* As column 0 of the matrix log of triangle A078812: */
%o {a(n)=local(A078812=matrix(n+1, n+1, r, c, if(r>=c, binomial(r+c-1,r-c))), LOG, ID=A078812^0); LOG=sum(m=1, n+1, -(ID-A078812)^m/m); n!*LOG[n+1, 1]}
%o (SageMath)
%o @CachedFunction
%o def T(n,k): # T = A123521
%o if (k==0): return 1
%o elif (k==1): return 2*(n-1)
%o else: return T(n-2, k-2) + binomial(2*n-k-1, 2*n-2*k-1)
%o @CachedFunction
%o def b(n):
%o if (n==1): return 1
%o else: return (-1/(2*(n-1)))*sum(T(n,j)*b(n-j+1) for j in (2..2*floor(n/2)))
%o def A179320(n): return 0 if (n==0) else 2*factorial(n)*b(n)
%o [A179320(n) for n in (0..40)] # _G. C. Greubel_, Sep 01 2022
%Y Cf. A027614, A078812, A179321, A261885.
%Y Variant: A179199.
%K sign
%O 0,2
%A _Paul D. Hanna_, Jul 11 2010
| 