A179320 E.g.f. satisfies: A(x) = A( x/(1-x)^2 ) * (1-x)/(1+x) with A(0)=0.
0, 2, -2, 6, -28, 160, -936, 4536, -20448, 627264, -19699200, 43908480, 17788273920, -211715112960, -41219197125120, 1301670191808000, 160057006683033600, -10037518414724505600, -1007362871616478003200
Offset: 0
Keywords
Examples
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! -+ ... 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! + ... where A(x/(1-x)^2) = (1+x)/(1-x)*A(x). ... Related expansions begin: . A = 2*x - 2*x^2/2! + 6*x^3/3! - 28*x^4/4! + 160*x^5/5! + ... . A*Dx(A)/2! = 8*x^2/2! - 30*x^3/3! + 180*x^4/4! - 1400*x^5/5! + ... . A*Dx(A*Dx(A))/3! = 48*x^3/3! - 416*x^4/4! + 4280*x^5/5! + ... . A*Dx(A*Dx(A*Dx(A)))/4! = 384*x^4/4! - 6160*x^5/5! + 98400*x^6/6! -+ ... . A*Dx(A*Dx(A*Dx(A*Dx(A))))/5! = 3840*x^5/5! - 100224*x^6/6! +- ... 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 + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 0..200
- J. Cigler, Some results and conjectures about a class of q-polynomials with simple moments, 2014.
Programs
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Mathematica
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 *) 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]}]]; A179320[n_] := 2*n!*b[n]; Table[A179320[n], {n, 0, 40}] (* G. C. Greubel, Sep 01 2022 *)
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PARI
/* E.g.f. satisfies: A(x) = (1-x)/(1+x)*A(x/(1-x)^2): */ {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)} -
PARI
/* 1/(1-x)^2 = 1 + A + A*Dx(A)/2! + A*Dx(A*Dx(A))/3! +...: */ {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)} -
PARI
/* As column 0 of the matrix log of triangle A078812: */ {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]}
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SageMath
@CachedFunction def T(n,k): # T = A123521 if (k==0): return 1 elif (k==1): return 2*(n-1) else: return T(n-2, k-2) + binomial(2*n-k-1, 2*n-2*k-1) @CachedFunction def b(n): if (n==1): return 1 else: return (-1/(2*(n-1)))*sum(T(n,j)*b(n-j+1) for j in (2..2*floor(n/2))) def A179320(n): return 0 if (n==0) else 2*factorial(n)*b(n) [A179320(n) for n in (0..40)] # G. C. Greubel, Sep 01 2022
Formula
E.g.f. A = A(x) satisfies:
(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! + ...
(2) Catalan(-x)^2 = 1 - A + A*Dx(A)/2! - A*Dx(A*Dx(A))/3! + A*Dx(A*Dx(A*Dx(A)))/4! -+ ...
(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! + ...
where Dx(F) = d/dx(x*F).
INVERSION FORMULA:
More generally, if A(x) = A(G(x))*G(x)/(x*G'(x)) with G(0)=0, G'(0)=1,
then G(x) can be obtained from A=A(x) by the series:
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).
ITERATION FORMULA:
Let G_{n}(x) denote the n-th iteration of G(x) = x/(1-x)^2, and A=A(x), then:
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! + ...
MATRIX LOG OF RIORDAN ARRAY (G(x)/x, G(x)) where G(x) = x/(1-x)^2:
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.
...
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