A231229 G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (2*k - x) / (1 - 2*k*x).

1, 2, 11, 96, 1117, 16226, 282503, 5732736, 132852169, 3461582882, 100170007859, 3187366835040, 110607201645061, 4157078373725762, 168223595493553871, 7292428324945552896, 337148168449629548497, 16559308764450704877506, 861070129828668874777883

Offset

0,2

Comments

Compare to a g.f. of Pell numbers (A000129):

Sum_{n>=0} x^n * Product_{k=1..n} (2*k + x)/(1 + 2*k*x) = 1/(1-2*x-x^2).

Limit n->infinity A231229(n) / A231277(n) = 2. - Vaclav Kotesovec, Nov 02 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..282

Formula

Self-convolution square-root yields A231277.

a(n) == (n+1) (mod 4).

a(n) ~ 2^(n-1) * n! / (log(2))^(n+1). - Vaclav Kotesovec, Nov 02 2014

Example

G.f.: A(x) = 1 + 2*x + 11*x^2 + 96*x^3 + 1117*x^4 + 16226*x^5 + 282503*x^6 +...
where
A(x) = 1 + x*(2-x)/(1-2*x) + x^2*(2-x)*(4-x)/((1-2*x)*(1-4*x)) + x^3*(2-x)*(4-x)*(6-x)/((1-2*x)*(1-4*x)*(1-6*x)) + x^4*(2-x)*(4-x)*(6-x)*(8-x)/((1-2*x)*(1-4*x)*(1-6*x)*(1-8*x)) +...
The g.f. equals the square of an integer series (A231277):
A(x)^(1/2) = 1 + x + 5*x^2 + 43*x^3 + 503*x^4 + 7395*x^5 + 130417*x^6 + 2677347*x^7 + 62652163*x^8 + 1645424927*x^9 + 47918249503*x^10 +...
because the g.f. A(x) is congruent to 1/(1-x)^2 modulo 4.

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, (2*k-x)/(1-2*k*x +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A231172, A231277.

Sequence in context: A332239 A261886 A245895 * A138210 A227465 A295099

Adjacent sequences: A231226 A231227 A231228 * A231230 A231231 A231232

Keyword

nonn

Author

Paul D. Hanna, Nov 05 2013

Status

approved