A231373 G.f. A(x) satisfies: A(x-x^2-x^3) = 1/sqrt(1-2*x-3*x^2), which is the g.f. the central trinomial coefficients (A002426).

1, 1, 4, 16, 71, 327, 1550, 7490, 36720, 182028, 910330, 4585318, 23233722, 118315318, 605088690, 3105994302, 15994906965, 82602799485, 427662046960, 2219130114108, 11538302709769, 60102637378353, 313591732265662, 1638671208390738, 8574718477933404, 44926247350136232

Offset: 0

Author

Paul D. Hanna, Nov 08 2013

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 4*x^2 + 16*x^3 + 71*x^4 + 327*x^5 + 1550*x^6 +...
where A(x-x^2-x^3)^2 = 1/(1-2*x-3*x^2):
A(x-x^2-x^3) = 1 + x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 141*x^6 + 393*x^7 + 1107*x^8 +...+ A002426(n)*x^n +...
The square of the g.f. begins (cf. A038112):
A(x)^2 = 1 + 2*x + 9*x^2 + 40*x^3 + 190*x^4 + 924*x^5 + 4578*x^6 +...
such that A(x)^2 = d/dx x*G(x) where G(x) is the g.f. of A001002:
G(x) = 1 + x + 3*x^2 + 10*x^3 + 38*x^4 + 154*x^5 + 654*x^6 +...
and satisfies G(x-x^2-x^3) = 1/(1-x-x^2).
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A002426, A038112, A001002.

Programs

Mathematica

CoefficientList[Series[Sqrt[D[InverseSeries[Series[x - x^2 - x^3, {x, 0, 30}], x], x]], {x, 0, 30}], x] (* Vaclav Kotesovec, Mar 31 2014 *)

PARI

{a(n)=local(G=serreverse(x-x^2-x^3+x^2*O(x^n)),A);A=sqrt(deriv(G));polcoeff(A,n)}
for(n=0,30,print1(a(n),", "))

PARI

{Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} \\ = d^n/dx^n F
{a(n)=local(A2=x); A2=1+sum(m=1, n+1, Dx(m, x^(2*m)*(1+x +x*O(x^n))^m/m!)); polcoeff(sqrt(A2), n)}
for(n=0,30,print1(a(n),", "))

Formula

Self-convolution yields A038112.

G.f. A(x) satisfies:

(1) A(x) = sqrt( Sum_{n>=0} d^n/dx^n x^(2*n)*(1+x)^n/n! ).

(2) A(x) = sqrt((1+x)*(5-27*x)*A(x)^6 - 1)/2, from a formula by Mark van Hoeij in A038112.

(3) A(x) = sqrt( d/dx x*G(x) ) where G(x) = Series_Reversion(x-x^2-x^3)/x is the g.f. of A001002.

(4) A(x) = 1/sqrt(1 - 2*x*G(x) - 3*x^2*G(x)^2) where G(x) = Series_Reversion(x-x^2-x^3)/x is the g.f. of A001002.

Sum_{k=0..n} a(k)*a(n-k) = Sum_{k=0..n} C(n+k, k)*C(k, n-k), from a formula by Paul Barry in A038112.

Recurrence: 25*(n-2)*(n-1)*n*a(n) = 110*(n-2)*(n-1)*(2*n-3)*a(n-1) - (n-2)*(214*n^2 - 856*n + 717)*a(n-2) - 33*(2*n-5)*(18*n^2 - 90*n + 113)*a(n-3) - 81*(n-3)*(3*n-11)*(3*n-7)*a(n-4). - Vaclav Kotesovec, Nov 10 2013

a(n) ~ 3^(3/4) * GAMMA(3/4) * (27/5)^n / (2*10^(1/4)*Pi*n^(3/4)). - Vaclav Kotesovec, Dec 29 2013