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

1,3

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)

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

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 *)

Prog

(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

Crossrefs

Cf. A123521, A179320.

Sequence in context: A218677 A353079 A305128 * A306040 A168592 A121873

Adjacent sequences: A027611 A027612 A027613 * A027615 A027616 A027617

Keyword

sign

Author

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

Status

approved