A027614 Related to Clebsch-Gordan formulas.

1, 1, 3, 14, 80, 468, 2268, 10224, 313632, 9849600, 21954240, -8894136960, -105857556480, 20609598562560, 650835095904000, -80028503341516800, -5018759207362252800, 503681435808239001600, 56090762228110443724800

Offset: 1

Author

Allan Adler (ara(AT)zurich.ai.mit.edu), Dec 15 1997

Keywords

• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 1..280
• Allan Adler, Definition (plain tex file)
• J. Cigler, Some results and conjectures about a class of q-polynomials with simple moments, 2014.
• Sean A. Irvine, Java program (github)

Crossrefs

Cf. A123521, A179320.

Programs

Mathematica

T[n_, k_]:= T[n, k]= If[k==0, 1, If[k==1, 2*(n-1), T[n-2, k-2] + Binomial[2*n-k-1, 2*n -2*k-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]}]];
A027614[n_]:= (-1)^(n+1)*n!*b[n];
Table[A027614[n], {n, 40}] (* G. C. Greubel, Sep 01 2022 *)

PARI

{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); (-1)^(n-1)*n!*polcoeff(A, n)/2};
vector(20, n, a(n)) \\ G. C. Greubel, Aug 23 2022

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 A027614(n): return (-1)^(n+1)*factorial(n)*b(n)
[A027614(n) for n in (1..40)] # G. C. Greubel, Sep 01 2022

Formula

a(n) = (-1)^(n-1)*A179320(n)/2. - G. C. Greubel, Aug 23 2022

a(n) = (-1)^(n+1) * n! * b(n), where b(n) = (-1/(2*(n-1))) * Sum_{j=2..2*floor(n/2)} A123521(n, j)*b(n-j+1), b(1) = 1. - G. C. Greubel, Sep 01 2022