OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..1173
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
FORMULA
a(n) = Sum_{k=0..n} C(2k+1,k)/(2k+1) * C(3n-k,n-k)*2k/(3n-k) for n>0 with a(0)=1.
G.f. satisfies: A(x) = 1 + x*G(x)^2*A(x)^2 where G(x) is the g.f. of A001764.
G.f. satisfies: A(x/F(x)) = F(x*F(x)) where F(x) is the g.f. of A000108 (Catalan).
From Alexander Burstein, Nov 23 2019: (Start)
G.f. satisfies: A(x) = 1 + x*G(x)^3*m(x*G(x)^3), where m(x) is the g.f. of A001006 (Motzkin numbers) and G(x) is the g.f. of A001764 (ternary trees).
G.f. satisfies: A(-x*A(x)^7) = 1/A(x). (End)
D-finite with recurrence: 16871976*(6*n + 5)*(3*n + 2)*(1 + 2*n)*(3*n + 1)*(6*n + 1)*a(n) + 891*(7899417*n^5 + 87634980*n^4 + 346831815*n^3 + 641723760*n^2 + 568823228*n + 195535920)*a(n + 1) + 198*(975573*n^5 + 86901660*n^4 + 832003665*n^3 + 3117626040*n^2 + 5226373402*n + 3282616420)*a(n + 2) - 24*(153776379*n^5 + 3051753345*n^4 + 23810889605*n^3 + 91660793285*n^2 + 174604614946*n + 131955815200)*a(n + 3) + 2*(660874241*n^5 + 14607391350*n^4 + 128653135535*n^3 + 564476075670*n^2 + 1234067785844*n + 1075696754160)*a(n + 4) - 180*(n + 6)*(979519*n^4 + 18386232*n^3 + 129147386*n^2 + 402350685*n + 469154448)*a(n + 5) + 1080*(2*n + 11)*(n + 7)*(n + 6)*(4132*n^2 + 37975*n + 88555)*a(n + 6) - 19440*(2*n + 11)*(n + 8)*(n + 7)*(n + 6)*(2*n + 13)*a(n + 7) = 0. - Robert Israel, Mar 18 2026
EXAMPLE
G.f.: A(x) = F(x*G(x)^2) = 1 + x + 4*x^2 + 20*x^3 + 110*x^4 +... where
F(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 +...
F(x)^2 = 1 + 2*x + 5*x^2 + 14*x^3 + 42*x^4 + 132*x^5 + 429*x^6 +...
G(x) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + 273*x^5 + 1428*x^6 +...
G(x)^2 = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 + 3876*x^6 +...
G(x)^3 = 1 + 3*x + 12*x^2 + 55*x^3 + 273*x^4 + 1428*x^5 +...
A(x)^2 = 1 + 2*x + 9*x^2 + 48*x^3 + 276*x^4 + 1656*x^5 +...
G(x)^2*A(x)^2 = 1 + 4*x + 20*x^2 + 110*x^3 + 638*x^4 +...
MAPLE
f:= gfun:-rectoproc({16871976*(6*n + 5)*(3*n + 2)*(1 + 2*n)*(3*n + 1)*(6*n + 1)*a(n) + 891*(7899417*n^5 + 87634980*n^4 + 346831815*n^3 + 641723760*n^2 + 568823228*n + 195535920)*a(n + 1) + 198*(975573*n^5 + 86901660*n^4 + 832003665*n^3 + 3117626040*n^2 + 5226373402*n + 3282616420)*a(n + 2) - 24*(153776379*n^5 + 3051753345*n^4 + 23810889605*n^3 + 91660793285*n^2 + 174604614946*n + 131955815200)*a(n + 3) + 2*(660874241*n^5 + 14607391350*n^4 + 128653135535*n^3 + 564476075670*n^2 + 1234067785844*n + 1075696754160)*a(n + 4) - 180*(n + 6)*(979519*n^4 + 18386232*n^3 + 129147386*n^2 + 402350685*n + 469154448)*a(n + 5) + 1080*(2*n + 11)*(n + 7)*(n + 6)*(4132*n^2 + 37975*n + 88555)*a(n + 6) - 19440*(2*n + 11)*(n + 8)*(n + 7)*(n + 6)*(2*n + 13)*a(n + 7), a(0) = 1, a(1) = 1, a(2) = 4, a(3) = 20, a(4) = 110, a(5) = 638, a(6) = 3828}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Mar 18 2026
PROG
(PARI) {a(n)=if(n==0, 1, sum(k=0, n, binomial(2*k+1, k)/(2*k+1)*binomial(3*(n-k)+2*k, n-k)*2*k/(3*(n-k)+2*k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 15 2009
STATUS
approved