A239453 Convolution of the generalized Catalan numbers A057977 with themselves.

1, 2, 3, 8, 11, 30, 43, 112, 172, 420, 694, 1584, 2809, 6006, 11379, 22880, 46088, 87516, 186562, 335920, 754646, 1293292, 3050238, 4992288, 12319816, 19315400, 49725004, 74884320, 200571541, 290845350, 808559299, 1131445440, 3257808976, 4407922860, 13119940234

Offset: 0

Author

Peter Luschny, Mar 19 2014

Keywords

• nonn

Crossrefs

Cf. A162551.

Programs

Sage

def A239453_list(n):
    r = sqrt(1-4*x^2)
    g = lambda x: (1-r)*(r+x)/(2*x^2*r)
    s = taylor(g(x), x, 0, n+1)
    f = [s.coefficient(x, j) for j in (0..n+1)]
    return [add(f[k]*f[j-k] for k in (0..j)) for j in (0..n)]
A239453_list(36)

Formula

D-finite with recurrence -(n+4)*(137*n-1035)*a(n) +(383*n^2-1622*n-5421)*a(n-1) +4*(274*n^2-1385*n-3534)*a(n-2) +4*(-766*n^2+4393*n+555)*a(n-3) +16*(-137*n^2+898*n-606)*a(n-4) +16*(n-3)*(383*n-1622)*a(n-5)=0. - R. J. Mathar, Feb 03 2025