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A222052 a(n) = A222051(n)/binomial(2*n,n), the central terms in rows of triangle A220178 divided by the central binomial coefficients. 2
1, 3, 25, 210, 1881, 17303, 162214, 1540710, 14776281, 142774455, 1387743525, 13553773500, 132906406950, 1307654814222, 12902933709922, 127632756058610, 1265251299930585, 12566655467547195, 125025126985317013, 1245750306517239978, 12429515281592007781 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..20.

FORMULA

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

a(n) = (2*n+1)*A222050(n), where g.f. G(x) of A222050 satisfies: G(x) = sqrt(1 + 2*x*G(x)^4 + 3*x^2*G(x)^6).

EXAMPLE

G.f.: A(x) = 1 + 3*x + 25*x^2 + 210*x^3 + 1881*x^4 + 17303*x^5 +...

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

Let G(x) = 1/sqrt(1-2*x-3*x^2) be 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 forms this sequence.

PROG

(PARI) {a(n)=polcoeff(1/sqrt(1-2*x-3*x^2+x*O(x^n))^(2*n+1), n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A222050, A222051.

Sequence in context: A037783 A037587 A280970 * A230718 A112240 A155640

Adjacent sequences:  A222049 A222050 A222051 * A222053 A222054 A222055

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 06 2013

STATUS

approved

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Last modified May 24 10:48 EDT 2019. Contains 323529 sequences. (Running on oeis4.)