A222050 G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^4 + 3*x^2*A(x)^6).

1, 1, 5, 30, 209, 1573, 12478, 102714, 869193, 7514445, 66083025, 589294500, 5316256278, 48431659786, 444928748618, 4117185679310, 38340948482745, 359047299072777, 3379057486089649, 31942315551724102, 303158909307122141, 2887629443604011421, 27595392738011189028

Offset

0,3

Links

Table of n, a(n) for n=0..22.

Formula

G.f.: sqrt( (1/x)*Series_Reversion( x*(1-2*x-3*x^2) ) ).

a(n) = [x^n] sqrt( 1/(1-2*x-3*x^2)^(2*n+1) ) / (2*n+1).

a(n) = A222052(n)/(2*n+1).

Example

G.f.: A(x) = 1 + x + 5*x^2 + 30*x^3 + 209*x^4 + 1573*x^5 + 12478*x^6 +...
Related expansions.
A(x)^2 = 1 + 2*x + 11*x^2 + 70*x^3 + 503*x^4 + 3864*x^5 + 31092*x^6 +...
A(x)^4 = 1 + 4*x + 26*x^2 + 184*x^3 + 1407*x^4 + 11280*x^5 + 93606*x^6 +...
A(x)^6 = 1 + 6*x + 45*x^2 + 350*x^3 + 2844*x^4 + 23814*x^5 + 204149*x^6 +...
where A(x)^2 = 1 + 2*x*A(x)^4 + 3*x^2*A(x)^6.
Let G(x) = 1/sqrt(1-2*x-3*x^2) denote the g.f. of A002426,
then the array of coefficients of x^k in G(x)^(2*n+1) begins:
G(x)^1 : [1,  1,   3,    7,    19,    51,    141,     393,...];
G(x)^3 : [1,  3,  12,   40,   135,   441,   1428,    4572,...];
G(x)^5 : [1,  5,  25,  105,   420,  1596,   5880,   21120,...];
G(x)^7 : [1,  7,  42,  210,   966,  4158,  17094,   67782,...];
G(x)^9 : [1,  9,  63,  363,  1881,  9009,  40755,  176319,...];
G(x)^11: [1, 11,  88,  572,  3289, 17303,  85228,  398684,...];
G(x)^13: [1, 13, 117,  845,  5330, 30498, 162214,  814606,...];
G(x)^15: [1, 15, 150, 1190,  8160, 50388, 287470, 1540710,...]; ...
in which the main diagonal (A222052) forms this sequence like so:
[1/1, 3/3, 25/5, 210/7, 1881/9, 17303/11, 162214/13, 1540710/15,...].

Prog

(PARI) {a(n)=polcoeff(sqrt(1/x*serreverse(x*(1-2*x-3*x^2)+x^2*O(x^n))), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(1/sqrt(1-2*x-3*x^2+x*O(x^n))^(2*n+1), n)/(2*n+1)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A222051, A222052, A002426.

Sequence in context: A367725 A082301 A144180 * A091122 A029587 A365178

Adjacent sequences: A222047 A222048 A222049 * A222051 A222052 A222053

Keyword

nonn

Author

Paul D. Hanna, Feb 06 2013

Status

approved