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A271469 G.f. satisfies: A(x) = 1 + x*(A(x)^3 - A(x)^4 + A(x)^5). 0
1, 1, 4, 23, 155, 1142, 8910, 72350, 605056, 5175866, 45077560, 398348733, 3562916317, 32192775763, 293410452560, 2694283228653, 24902681767987, 231496130358758, 2162985033344112, 20301976721356134, 191336242071696514, 1809916398759630481, 17178063381786563194, 163536967014934201972, 1561247114394683682834, 14943175106109268856975 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..25.

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

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

Cf. A219537, A001006.

Sequence in context: A192730 A246813 A055723 * A007297 A263843 A326350

Adjacent sequences: A271466 A271467 A271468 * A271470 A271471 A271472

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Apr 08 2016

STATUS

approved

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Last modified December 5 04:50 EST 2022. Contains 358578 sequences. (Running on oeis4.)