A064087 - OEIS

%I #28 Jul 06 2021 01:57:38

%S 1,1,5,41,413,4641,55797,702297,9137549,121909457,1658755685,

%T 22929591433,321111942781,4546112358529,64958195967957,

%U 935566629270201,13567825195172973,197957440018622769,2903721563443327557,42796201522669935081,633443408407612143453

%N Generalized Catalan numbers C(4; n).

%C a(n+1) = Y_{n}(n+1) = Z_{n} in the Derrida et al. 1992 reference (see A064094) for alpha=4, beta=1 (or alpha=1, beta=4).

%F G.f.: (1+4*x*c(4*x)/3)/(1+x/3) = 1/(1-x*c(4*x)) with c(x) g.f. of Catalan numbers A000108.

%F a(n) = (1/n)*Sum_{m=0..n-1} (n-m)*binomial(n-1+m, m)*(4^m) = ((-1/3)^n)*(1 - 4*Sum_{k=0..n-1} C(k)*(-12)^k), n >= 1, a(0) = 1, with C(n) = A000108(n) (Catalan).

%F a(n) = Sum_{k=0...n} A059365(n, k)*4^(n-k). - _Philippe Deléham_, Jan 19 2004

%F D-finite with recurrence: 3*n*a(n) + (-47*n+72)*a(n-1) + 8*(-2*n+3)*a(n-2) = 0. - _R. J. Mathar_, Jun 07 2013 [verified by _Georg Fischer_, Jul 06 2021]

%F a(n) = hypergeometric([1-n, n], [-n], 4) for n > 0. - _Peter Luschny_, Nov 30 2014

%F a(n) ~ 2^(4*n + 2) / (49*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 10 2019

%t a[0] = 1; a[n_] := Sum[(n - m)*Binomial[n - 1 + m, m]*4^m/n, {m, 0, n - 1}]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Jul 09 2013 *)

%o (PARI)

%o a(n) = if(n<0,0,polcoeff(serreverse((x-3*x^2)/(1+x)^2+O(x^(n+1))),n)) /* _Ralf Stephan_ */

%o (Sage)

%o def a(n):

%o     if n==0: return 1

%o     return hypergeometric([1-n, n], [-n], 4).simplify()

%o [a(n) for n in range(24)] # _Peter Luschny_, Nov 30 2014

%Y Cf. A064063 (C(3; n)).

%K nonn,easy

%O 0,3

%A _Wolfdieter Lang_, Sep 13 2001