A206847 a(n) = Sum_{k=0..n} binomial(n^2, k^2) * binomial(n^2, (n-k)^2).

1, 2, 18, 2270, 3678482, 51789416252, 9723940840418814, 13783866167176942874214, 260749663122506218247699587346, 35385577627626083328957267246097557212, 64138056102285851525440919122006580387539950268, 814449089808478655249485968539593253265395820497817710866

Offset

0,2

Comments

Ignoring the initial term a(0), equals the logarithmic derivative of A206846.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..45

Formula

From Vaclav Kotesovec, Mar 03 2014: (Start)

Limit n->infinity a(n)^(1/n^2) = 16/(3*sqrt(3)).

a(n) ~ c * 2^(4*n^2+3) / (Pi * n^2 * 3^(3*n^2/2+1)), where c = JacobiTheta3(0,9*exp(-16/3)) = EllipticTheta[3, 0, 9*Exp[-16/3]] = 1.08691022925895131... if n is even, and c = JacobiTheta2(0,9*exp(-16/3)) = EllipticTheta[2, 0, 9*Exp[-16/3]] = 0.91485129628884995... if n is odd.

(End)

Example

L.g.f.: L(x) = 2*x + 18*x^2/2 + 2270*x^3/3 + 3678482*x^4/4 + 51789416252*x^5/5 +...
where exponentiation yields the g.f. of A206846:
exp(L(x)) = 1 + 2*x + 11*x^2 + 776*x^3 + 921193*x^4 + 10359730908*x^5 +...
Illustration of terms: by definition,
a(1) = C(1,0)*C(1,1) + C(1,1)*C(1,0);
a(2) = C(4,0)*C(4,4) + C(4,1)*C(4,1) + C(4,4)*C(4,0);
a(3) = C(9,0)*C(9,9) + C(9,1)*C(9,4) + C(9,4)*C(9,1) + C(9,9)*C(9,0);
a(4) = C(16,0)*C(16,16) + C(16,1)*C(16,9) + C(16,4)*C(16,4) + C(16,9)*C(16,1) + C(16,16)*C(16,0); ...
Numerically, the above evaluates to be:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 4*4 + 1*1 = 18;
a(3) = 1*1 + 9*126 + 126*9 + 1*1 = 2270;
a(4) = 1*1 + 16*11440 + 1820*1820 + 11440*16 + 1*1 = 3678482;
a(5) = 1*1 + 25*2042975 + 12650*2042975 + 2042975*12650 + 2042975*25 + 1*1 = 51789416252; ...

Mathematica

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

Prog

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

Crossrefs

Cf. A206846 (exp), A206849, A206851.

Sequence in context: A260610 A333164 A076954 * A259654 A060598 A055687

Adjacent sequences: A206844 A206845 A206846 * A206848 A206849 A206850

Keyword

nonn

Author

Paul D. Hanna, Feb 15 2012

Status

approved