A201169 G.f. A(x) satisfies A(x) = 1 + x/sqrt(1-4*x^2) * A(x/sqrt(1-4*x^2)).

1, 1, 1, 3, 7, 19, 59, 179, 615, 2141, 7853, 30179, 118663, 487581, 2050325, 8890743, 39593811, 180367655, 842738375, 4018170415, 19578229355, 97275623919, 492406520567, 2538806377419, 13312298342703, 70986151599775, 384532625928335, 2115147900599015

Offset

0,4

Links

Table of n, a(n) for n=0..27.

Formula

a(n) = sum(m=0..(n-1)/2, a(n-2*m-1) * binomial(n/2-1,m) * 2^(2*m)), n>0, a(0)=1.

G.f.: Sum_{n>=0} x^n / Product_{k=0..n} sqrt(1 - 4*k*x^2). - Paul D. Hanna, Nov 30 2014

G.f. A(x) satisfies: A(x/sqrt(1+4*x^2)) = 1 + x*A(x). - Paul D. Hanna, Nov 30 2014

Example

G.f.: A(x) = = 1 + x + x^2 + 3*x^3 + 7*x^4 + 19*x^5 + 59*x^6 + 179*x^7 + ...
such that A(x) = 1 + G(x)*A(G(x)) where
G(x) = x/sqrt(1-4*x^2) = x + 2*x^3 + 6*x^5 + 20*x^7 + 70*x^9 + 252*x^11 + ... .
The g.f. also equals the series
A(x) = 1 + x/sqrt(1-4*x^2) + x^2/sqrt((1-4*x^2)*(1-8*x^2)) + x^3/sqrt((1-4*x^2)*(1-8*x^2)*(1-12*x^2)) + x^4/sqrt((1-4*x^2)*(1-8*x^2)*(1-12*x^2)*(1-16*x^2)) + ... .

Prog

(Maxima)
a(n):=if n=0 then 1 else sum(a(n-2*m-1)*binomial(n/2-1, m)*2^(2*m), m, 0, (n-1)/2);
(PARI) {a(n)=polcoeff( sum(m=0, n, x^m / prod(k=0, m, sqrt(1 - 4*k*x^2 +x*O(x^n)) )), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 30 2014

Crossrefs

Sequence in context: A005506 A268125 A287484 * A148667 A148668 A148669

Adjacent sequences: A201166 A201167 A201168 * A201170 A201171 A201172

Keyword

nonn

Author

Vladimir Kruchinin, Nov 28 2011

Extensions

Name corrected slightly by Paul D. Hanna, Nov 30 2014

Status

approved