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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
OFFSET
0,2
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
Sequence in context: A037783 A037587 A280970 * A230718 A112240 A155640
KEYWORD
nonn,changed
AUTHOR
Paul D. Hanna, Feb 06 2013
STATUS
approved