A168452 - OEIS

%I #12 Jan 22 2017 15:24:45

%S 1,4,24,180,1556,14840,152092,1646652,18613664,217852008,2623657384,

%T 32361812912,407342311632,5217211974832,67836910362772,

%U 893766246630572,11913422912188432,160450066324972472,2181014117345997704,29894260817385950064,412839378639052110464

%N Self-convolution of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).

%H Alois P. Heinz, <a href="/A168452/b168452.txt">Table of n, a(n) for n = 0..500</a>

%F G.f.: A(x) = (1/x)*Series_Reversion(x/G(x)^2) where G(x) = g.f. of A004304, where A004304(n) is the number of planar tree-rooted maps with n edges.

%F G.f.: A(x) = G(x*A(x))^2 where A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A004304.

%F a(n) ~ c * 16^n / n^3, where c = 3.07968404... . - _Vaclav Kotesovec_, Sep 12 2014

%e G.f.: A(x) = 1 + 4*x + 24*x^2 + 180*x^3 + 1556*x^4 + 14840*x^5 +...

%e A(x)^(1/2) = 1 + 2*x + 10*x^2 + 70*x^3 + 588*x^4 + 5544*x^5 + 56628*x^6 +...+ A005568(n)*x^n +...

%e A(x) satisfies: A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A004304:

%e G(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 28*x^4 + 160*x^5 + 1036*x^6 +...+ A004304(n)*x^n +...

%e G(x)^2 = 1 + 4*x + 8*x^2 + 20*x^3 + 84*x^4 + 456*x^5 + 2860*x^6 +...+ A168451(n)*x^n +...

%p a:= proc(n) option remember; `if`(n<3, [1, 4, 24][n+1],

%p       (12*n*(n+1)*(16*n^4+68*n^3+44*n^2-63*n-25) *a(n-1)

%p        -(3072*n^6+768*n^5-8448*n^4+1152*n^3+3264*n^2-288) *a(n-2)

%p        +1024*n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(4*n+1) *a(n-3)) /

%p       ((n+1)^2*(n+2)*(n+3)*(n+4)*(4*n-3)))

%p     end:

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 20 2013

%t c[n_] := CatalanNumber[n]*CatalanNumber[n+1]; a[n_] := ListConvolve[cc = Array[c, n+1, 0], cc][[1]]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Jan 22 2017 *)

%o (PARI) {a(n)=local(C_2=vector(n+1,m,(binomial(2*m-2,m-1)/m)*(binomial(2*m,m)/(m+1))));polcoeff(Ser(C_2)^2,n)}

%Y Cf. A168451, A004304, A005568, A000108, variant: A168358.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 26 2009

%E Typo in formula corrected by _Paul D. Hanna_, Nov 28 2009