A168358 -id:A168358 - 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).

1, 4, 24, 180, 1556, 14840, 152092, 1646652, 18613664, 217852008, 2623657384, 32361812912, 407342311632, 5217211974832, 67836910362772, 893766246630572, 11913422912188432, 160450066324972472, 2181014117345997704, 29894260817385950064, 412839378639052110464

Offset: 0

Paul D. Hanna, Nov 26 2009

nonn

			G.f.: A(x) = 1 + 4*x + 24*x^2 + 180*x^3 + 1556*x^4 + 14840*x^5 +...
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 +...
A(x) satisfies: A(x/G(x)^2) = G(x)^2 where G(x) = g.f. of A004304:
G(x) = 1 + 2*x + 2*x^2 + 6*x^3 + 28*x^4 + 160*x^5 + 1036*x^6 +...+ A004304(n)*x^n +...
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 +...

Alois P. Heinz, Table of n, a(n) for n = 0..500

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

a:= proc(n) option remember; `if`(n<3, [1, 4, 24][n+1],
      (12*n*(n+1)*(16*n^4+68*n^3+44*n^2-63*n-25) *a(n-1)
       -(3072*n^6+768*n^5-8448*n^4+1152*n^3+3264*n^2-288) *a(n-2)
       +1024*n*(n-1)*(n-2)*(2*n-1)*(2*n-3)*(4*n+1) *a(n-3)) /
      ((n+1)^2*(n+2)*(n+3)*(n+4)*(4*n-3)))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 20 2013

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 *)

{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)}

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.
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.
a(n) ~ c * 16^n / n^3, where c = 3.07968404... . - Vaclav Kotesovec, Sep 12 2014
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

Typo in formula corrected by Paul D. Hanna, Nov 28 2009

A168357 Self-convolution of A006664, which is the number of irreducible systems of meanders.

Original entry on oeis.org

1, 2, 5, 20, 112, 768, 5984, 50856, 460180, 4366076, 42988488, 436066232, 4532973676, 48095557700, 519247705968, 5690272928520, 63172884082028, 709373555125356, 8046263496489260, 92089662771965492, 1062482514810065752

Offset: 0

Paul D. Hanna, Nov 23 2009

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 20*x^3 + 112*x^4 + 768*x^5 +...
A(x)^(1/2) = 1 + x + 2*x^2 + 8*x^3 + 46*x^4 + 322*x^5 + 2546*x^6 +...+ A006664(n)*x^n +...
G.f. satisfies: A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A001246:
F(x) = 1 + x + 4*x^2 + 25*x^3 + 196*x^4 + 1764*x^5 + 17424*x^6 +...+ A000108(n)^2*x^n +...
F(x)^2 = 1 + 2*x + 9*x^2 + 58*x^3 + 458*x^4 + 4120*x^5 + 40569*x^6 +...+ A168358(n)*x^n +...

Cf. A168358, A168344, A001246, A006664, A000108.

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

G.f.: A(x) = x/Series_Reversion(x*F(x)^2) where F(x) = g.f. of A001246, which is the squares of Catalan numbers.

G.f.: A(x) = F(x/A(x))^2 where A(x*F(x)^2) = F(x)^2 where F(x) = g.f. of A001246.

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).

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