A206851 L.g.f.: Sum_{n>=1} x^n/n * Sum_{k=0..n} binomial(n^2, k^2)*x^k = Sum_{n>=1} a(n)*x^n/n.

1, 3, 7, 15, 231, 2763, 37773, 3347359, 145164760, 15115517783, 5300285945494, 841490209145991, 700215432847179640, 821522962294608211319, 580955012898669141073842, 3240132942509582109732641935, 12114306457535986210506222037102

Offset: 1

Author

Paul D. Hanna, Feb 15 2012

Keywords

• nonn

Comments

Equals the logarithmic derivative of A206850.

Examples

			L.g.f.: L(x) = x + 3*x^2/2 + 7*x^3/3 + 15*x^4/4 + 231*x^5/5 + 2763*x^6/6 +...
where exponentiation yields the g.f. of A206850:
exp(L(x)) = 1 + x + 2*x^2 + 4*x^3 + 8*x^4 + 56*x^5 + 522*x^6 + 5972*x^7 +...
By definition, the l.g.f. equals the series:
L(x) = (C(1,0) + C(1,1)*x)*x
+ (C(4,0) + C(4,1)*x + C(4,4)*x^2)*x^2/2
+ (C(9,0) + C(9,1)*x + C(9,4)*x^2 + C(9,9)*x^3)*x^3/3
+ (C(16,0) + C(16,1)*x + C(16,4)*x^2 + C(16,9)*x^3 + C(16,16)*x^4)*x^4/4
+ (C(25,0) + C(25,1)*x + C(25,4)*x^2 + C(25,9)*x^3 + C(25,16)*x^4 + C(25,25)*x^5)*x^5/5 +...
More explicitly,
L(x) = (1 + 1*x)*x + (1 + 4*x + 1*x^2)*x^2/2
+ (1 + 9*x + 126*x^2 + 1*x^3)*x^3/3
+ (1 + 16*x + 1820*x^2 + 11440*x^3 + 1*x^4)*x^4/4
+ (1 + 25*x + 12650*x^2 + 2042975*x^3 + 2042975*x^4 + 1*x^5)*x^5/5
+ (1 + 36*x + 58905*x^2 + 94143280*x^3 + 7307872110*x^4 + 600805296*x^5 + 1*x^6)*x^6/6 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..94

Crossrefs

Cf. A206850 (exp), A206847, A206849, A123165.

Programs

Mathematica

Table[n*Sum[Binomial[(n-k)^2, k^2]/(n-k),{k,0,Floor[n/2]}], {n, 1, 20}] (* Vaclav Kotesovec, Mar 03 2014 *)

PARI

{a(n)=n*polcoeff(sum(m=1,n,x^m/m*sum(k=0,m,binomial(m^2,k^2)*x^k)+x*O(x^n)),n)}

PARI

{a(n)=n*sum(k=0,n\2, binomial((n-k)^2, k^2)/(n-k))}
for(n=1, 20, print1(a(n), ", "))

Formula

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

Limit n->infinity a(n)^(1/n^2) = (1-2*r)^r / r^(2*r) = 1.2915356633069917227119166349..., where r = A323778 = 0.365498498219858044579... is the root of the equation (1-r)^(2-2*r) * r^(2*r) = 1-2*r. - Vaclav Kotesovec, Mar 03 2014