A133602 - OEIS

%I #15 Jun 06 2020 15:32:32

%S 1,2,2,7,14,56,132,561,1430,6292,16796,75582,208012,950912,2674440,

%T 12369285,35357670,165002460,477638700,2244901890,6564120420,

%U 31030387440,91482563640,434542177290,1289904147324,6151850548776

%N The matrix-vector product A133080 * A000108.

%C A133603 is a companion sequence.

%F A133080 * A000108, where A133080 = an infinite lower triangular matrix and A000108 = the Catalan sequence as a vector.

%F a(2n) = A048990(n).

%F a(2n+1) = A005807(2n).

%F Conjecture: n*(n-1)*(n-3)*(3*n-4)*a(n) -8*(n-1)*(2*n-5)*a(n-1) -4*(n-2)*(3*n-1)*(2*n-5)*(2*n-7)*a(n-2)=0. - _R. J. Mathar_, Jun 20 2015

%e a(4) = C(4) = 14.

%e a(5) = 56 = C(5) + C(4) = 42 + 14.

%o (Python)

%o from sympy import catalan

%o def a005807(n): return catalan(n) + catalan(n + 1)

%o def a048990(n): return catalan(2*n)

%o l=[1, 2]

%o for n in range(2, 31): l+=[a048990(n//2) if n%2==0 else a005807(n - 1)]

%o print(l) # _Indranil Ghosh_, Jul 15 2017

%Y Cf. A097806, A133080, A000108, A133603.

%K nonn,easy

%O 0,2

%A _Gary W. Adamson_, Sep 18 2007