A027614 - OEIS

%I #20 Sep 03 2022 10:44:24

%S 1,1,3,14,80,468,2268,10224,313632,9849600,21954240,-8894136960,

%T -105857556480,20609598562560,650835095904000,-80028503341516800,

%U -5018759207362252800,503681435808239001600,56090762228110443724800

%N Related to Clebsch-Gordan formulas.

%H G. C. Greubel, <a href="/A027614/b027614.txt">Table of n, a(n) for n = 1..280</a>

%H Allan Adler, <a href="/A027614/a027614.tex">Definition (plain tex file)</a>

%H J. Cigler, <a href="http://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.

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a027/A027614.java">Java program</a> (github)

%F a(n) = (-1)^(n-1)*A179320(n)/2. - _G. C. Greubel_, Aug 23 2022

%F 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

%t 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 *)

%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 A027614[n_]:= (-1)^(n+1)*n!*b[n];

%t Table[A027614[n], {n, 40}] (* _G. C. Greubel_, Sep 01 2022 *)

%o (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};

%o vector(20, n, a(n)) \\ _G. C. Greubel_, Aug 23 2022

%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 A027614(n): return (-1)^(n+1)*factorial(n)*b(n)

%o [A027614(n) for n in (1..40)] # _G. C. Greubel_, Sep 01 2022

%Y Cf. A123521, A179320.

%K sign

%O 1,3

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