A207138 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k*(n-k))*x^k = Sum_{n>=1} a(n)*x^n/n.

1, 3, 7, 51, 761, 17913, 688745, 56611987, 11405877739, 4272862207703, 2450039810788461, 2224842228379519641, 4169966883810355864393, 19139862395982576668262825, 166161479603614500915921996017, 2206856314384330228779059994929555

Offset

1,2

Comments

Equals the logarithmic derivative of A207137.

Links

Table of n, a(n) for n=1..16.

Formula

a(n) = n*Sum_{k=0..[n/2]} binomial((n-k)^2, k*(n-2*k))/(n-k).

Limit n->infinity a(n)^(1/n^2) = ((1-r)^2/(r*(1-2*r)))^((1-3*r)*(1-r)/(3*(1-2*r))) = 1.36198508972775011599..., where r = 0.195220321930105755... is the root of the equation (1-3*r+3*r^2)^(3*(2*r-1)) = (r*(1-2*r))^(4*r-1) * (1-r)^(4*(r-1)). - Vaclav Kotesovec, Mar 04 2014

Example

L.g.f.: L(x) =  x + 3*x^2/2 + 7*x^3/3 + 51*x^4/4 + 761*x^5/5 + 17913*x^6/6 +...
where exponentiation equals the g.f. of A207137:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 17*x^4 + 171*x^5 + 3171*x^6 +...
To illustrate the definition, the l.g.f. equals the series:
L(x) = (1 + x)*x + (1 + 4*x + 1*x^2)*x^2/2
+ (1 + 36*x + 36*x^2 + 1*x^3)*x^3/3
+ (1 + 560*x + 1820*x^2 + 560*x^3 + 1*x^4)*x^4/4
+ (1 + 12650*x + 177100*x^2 + 177100*x^3 + 12650*x^4 + 1*x^5)*x^5/5
+ (1 + 376992*x + 30260340*x^2 + 94143280*x^3 + 30260340*x^4 + 376992*x^5 + 1*x^6)*x^6/6 +...

Mathematica

Table[n*Sum[Binomial[(n-k)^2, k*(n-2*k)]/(n-k), {k, 0, Floor[n/2]}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 04 2014 *)

Prog

(PARI) {a(n)=n*polcoeff(sum(m=1, n+1, x^m/m*sum(k=0, m, binomial(m^2, k*(m-k))*x^k))+x*O(x^n), n)}
(PARI) {a(n)=n*sum(k=0, n\2, binomial((n-k)^2, k*(n-2*k))/(n-k))}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A207137 (exp), A207136, A167006, A228837.

Sequence in context: A273092 A120788 A346143 * A041277 A248239 A239317

Adjacent sequences: A207135 A207136 A207137 * A207139 A207140 A207141

Keyword

nonn

Author

Paul D. Hanna, Feb 15 2012

Status

approved