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

0,3

Links

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

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

Example

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).

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 *)

Prog

(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), ", "))

Crossrefs

Cf. A002426, A038112, A001002.

Sequence in context: A326654 A246014 A373959 * A161797 A124533 A227034

Adjacent sequences: A231370 A231371 A231372 * A231374 A231375 A231376

Keyword

nonn

Author

Paul D. Hanna, Nov 08 2013

Status

approved