OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..998
FORMULA
G.f. A(x) satisfies:
(1) A(x)^2 = 1 + x*(A(x)^3 + A(x)^6).
(2) A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7).
Let F(x) = (1+x - sqrt(1 - 2*x - 3*x^2)) / (2*x), then g.f. A(x) satisfies:
(3) A(x) = ( (1/x)*Series_Reversion(x/F(x)^3) )^(1/3),
(4) A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3),
where F(x) = 1 + x*M(x) such that M(x) = 1 + x*M(x) + x^2*M(x)^2 is the g.f. of the Motzkin numbers (A001006).
Let G(x) = 1 + x*(G(x)^2 - G(x)^3 + G(x)^4), then g.f. A(x) satisfies:
(5) A(x) = (1/x)*Series_Reversion(x/G(x)),
(6) A(x) = G(x*A(x)) and G(x) = A(x/G(x)),
where G(x) is the g.f. of A219537.
a(n) ~ sqrt((34 + (34102 - 8262*sqrt(17))^(1/3) + (34102 + 8262*sqrt(17))^(1/3)) / 1632) * ((28 + (513243 - 4131*sqrt(17))^(1/3)/3 + (19009 + 153*sqrt(17))^(1/3)) / 8)^n / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Apr 16 2016
D-finite recurrence: 8*n*(2*n-1)*(4*n-1)*(4*n+1)*(204*n^4 - 1341*n^3 + 3191*n^2 - 3286*n + 1242)*a(n) = 12*(45696*n^8 - 391776*n^7 + 1376164*n^6 - 2580579*n^5 + 2808064*n^4 - 1797694*n^3 + 651566*n^2 - 119476*n + 8160)*a(n-1) - 6*(n-2)*(29376*n^7 - 237168*n^6 + 760044*n^5 - 1236774*n^4 + 1082233*n^3 - 496791*n^2 + 108530*n - 8400)*a(n-2) + 9*(n-3)*(n-2)*(3*n-8)*(3*n-4)*(204*n^4 - 525*n^3 + 392*n^2 - 111*n + 10)*a(n-3). - Vaclav Kotesovec, Apr 16 2016
From Seiichi Manyama, Aug 06 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} binomial(n,k) * binomial(3*n+k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * binomial(n,k) * binomial(5*n-2*k,n-1-k) for n > 0. (End)
G.f.: A(x) = sqrt(B(x)) where B(x) is the g.f. of A370474. - Seiichi Manyama, Mar 31 2024
a(n) = (1/n) * Sum_{k=0..floor(n-1)/2} binomial(n,k) * binomial(4*n-k,n-1-2*k) for n > 0. - Seiichi Manyama, Apr 01 2024
a(n) = Sum_{k=0..n} binomial(n,k) * binomial(3*n/2+3*k/2+1/2,n)/(3*n+3*k+1). - Seiichi Manyama, Apr 04 2024
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 155*x^4 + 1142*x^5 + 8910*x^6 +...
Related expansions:
A(x)^2 = 1 + 2*x + 9*x^2 + 54*x^3 + 372*x^4 + 2778*x^5 + 21873*x^6 +...
A(x)^3 = 1 + 3*x + 15*x^2 + 94*x^3 + 663*x^4 + 5025*x^5 + 39970*x^6 +...
A(x)^4 = 1 + 4*x + 22*x^2 + 144*x^3 + 1041*x^4 + 8016*x^5 + 64470*x^6 +...
A(x)^5 = 1 + 5*x + 30*x^2 + 205*x^3 + 1520*x^4 + 11901*x^5 + 96850*x^6 +...
A(x)^6 = 1 + 6*x + 39*x^2 + 278*x^3 + 2115*x^4 + 16848*x^5 + 138816*x^6 +...
A(x)^7 = 1 + 7*x + 49*x^2 + 364*x^3 + 2842*x^4 + 23044*x^5 + 192325*x^6 +...
where A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5),
and A(x)^2 = 1 + x*(A(x)^3 + A(x)^6),
and A(x)^3 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^7),
and A(x)^4 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^8),
and A(x)^5 = 1 + x*(A(x)^3 + A(x)^5 + A(x)^6 + A(x)^7 + A(x)^9), etc.
The g.f. satisfies A(x) = F(x*A(x)^3) and F(x) = A(x/F(x)^3) where
F(x) = 1 + x + x^2 + 2*x^3 + 4*x^4 + 9*x^5 + 21*x^6 + 51*x^7 +...+ A001006(n-1)*x^n +...
is a g.f. of the Motzkin numbers (A001006, shifted right 1 place).
The g.f. satisfies A(x) = G(x*A(x)) and G(x) = A(x/G(x)) where
G(x) = 1 + x + 3*x^2 + 13*x^3 + 66*x^4 + 366*x^5 + 2148*x^6 +...+ A219537(n)*x^n +...
satisfies G(x) = 1 + x*(G(x)^2 - G(x)^3 + G(x)^4).
MATHEMATICA
CoefficientList[(1/x*InverseSeries[Series[8*x^4/(1 + x - Sqrt[1 - 2*x - 3*x^2])^3, {x, 0, 20}], x])^(1/3), x] (* Vaclav Kotesovec, Apr 16 2016 *)
PROG
(PARI) /* Formula A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5): */
{a(n)=local(A=1); for(i=1, n, A=1+x*(A^3-A^4+A^5) +x*O(x^n)); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Formula using Series Reversion involving Motzkin numbers: */
{a(n)=local(A=1); A=(1+x-sqrt(1-2*x-3*x^2+x^3*O(x^n)))/(2*x); polcoeff( (1/x*serreverse(x/A^3))^(1/3), n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 08 2016
STATUS
approved