cp's OEIS Frontend

A168452 Self-convolution of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).

%I A168452 #14 Nov 22 2024 06:23:47
%S A168452 1,4,24,180,1556,14840,152092,1646652,18613664,217852008,2623657384,
%T A168452 32361812912,407342311632,5217211974832,67836910362772,
%U A168452 893766246630572,11913422912188432,160450066324972472,2181014117345997704,29894260817385950064,412839378639052110464
%N A168452 Self-convolution of A005568, where A005568(n) is the product of two successive Catalan numbers C(n)*C(n+1).
%H A168452 Alois P. Heinz, <a href="/A168452/b168452.txt">Table of n, a(n) for n = 0..500</a>
%F A168452 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 A168452 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 A168452 a(n) ~ c * 16^n / n^3, where c = 3.07968404... . - _Vaclav Kotesovec_, Sep 12 2014
%F A168452 Conjecture D-finite with recurrence 3*(n+4)*(n+3)*(n+2)*(n+1)^2*a(n) -4*n*(n+1) *(32*n^3+164*n^2+233*n+75)*a(n-1) +96*(16*n^5+24*n^4-14*n^3-28*n^2-16*n+3) *a(n-2) +1536*(-8*n^4+22*n^3-32*n+15)*a(n-3) -16384*(2*n-5)*(n-1)*(n-2)*(n-3)*(2*n-3)*a(n-4)=0. - _R. J. Mathar_, Nov 22 2024
%e A168452 G.f.: A(x) = 1 + 4*x + 24*x^2 + 180*x^3 + 1556*x^4 + 14840*x^5 +...
%e A168452 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 A168452 A(x) satisfies: A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A004304:
%e A168452 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 A168452 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 A168452 a:= proc(n) option remember; `if`(n<3, [1, 4, 24][n+1],
%p A168452       (12*n*(n+1)*(16*n^4+68*n^3+44*n^2-63*n-25) *a(n-1)
%p A168452        -(3072*n^6+768*n^5-8448*n^4+1152*n^3+3264*n^2-288) *a(n-2)
%p A168452        +1024*n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(4*n+1) *a(n-3)) /
%p A168452       ((n+1)^2*(n+2)*(n+3)*(n+4)*(4*n-3)))
%p A168452     end:
%p A168452 seq(a(n), n=0..25);  # _Alois P. Heinz_, Oct 20 2013
%t A168452 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 A168452 (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 A168452 Cf. A168451, A004304, A005568, A000108, variant: A168358.
%K A168452 nonn
%O A168452 0,2
%A A168452 _Paul D. Hanna_, Nov 26 2009
%E A168452 Typo in formula corrected by _Paul D. Hanna_, Nov 28 2009