A206849 a(n) = Sum_{k=0..n} binomial(n^2, k^2).

1, 2, 6, 137, 13278, 4098627, 8002879629, 66818063663192, 1520456935214867934, 167021181249536494996841, 102867734705055054467692090431, 179314863425920182637610314008444247, 1094998941099523423274757578750950802034789

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..57

Vaclav Kotesovec, Limits, graph for 500 terms

Formula

Ignoring the initial term a(0), equals the logarithmic derivative of A206848.

Equals the row sums of triangle A226234.

From Vaclav Kotesovec, Mar 03 2014: (Start)

Limit n->infinity a(n)^(1/n^2) = 2

Lim sup n->infinity a(n)/(2^(n^2)/n) = sqrt(2/Pi) * JacobiTheta3(0,exp(-4)) = Sqrt[2/Pi] * EllipticTheta[3, 0, 1/E^4] = 0.827112271364145742...

Lim inf n->infinity a(n)/(2^(n^2)/n) = sqrt(2/Pi) * JacobiTheta2(0,exp(-4)) = Sqrt[2/Pi] * EllipticTheta[2, 0, 1/E^4] = 0.587247586271786487...

(End)

Example

L.g.f.: L(x) = 2*x + 6*x^2/2 + 137*x^3/3 + 13278*x^4/4 + 4098627*x^5/5 +...
where exponentiation yields the g.f. of A206848:
exp(L(x)) = 1 + 2*x + 5*x^2 + 53*x^3 + 3422*x^4 + 826606*x^5 + 1335470713*x^6 +...
Illustration of terms: by definition,
a(1) = C(1,0) + C(1,1);
a(2) = C(4,0) + C(4,1) + C(4,4);
a(3) = C(9,0) + C(9,1) + C(9,4) + C(9,9);
a(4) = C(16,0) + C(16,1) + C(16,4) + C(16,9) + C(16,16);
a(5) = C(25,0) + C(25,1) + C(25,4) + C(25,9) + C(25,16) + C(25,25);
a(6) = C(36,0) + C(36,1) + C(36,4) + C(36,9) + C(36,16) + C(36,25) + C(36,36); ...
Numerically, the above evaluates to be:
a(1) = 1 + 1 = 2;
a(2) = 1 + 4 + 1 = 6;
a(3) = 1 + 9 + 126 + 1 = 137;
a(4) = 1 + 16 + 1820 + 11440 + 1 = 13278;
a(5) = 1 + 25 + 12650 + 2042975 + 2042975 + 1 = 4098627;
a(6) = 1 + 36 + 58905 + 94143280 + 7307872110 + 600805296 + 1 = 8002879629; ...

Mathematica

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

Prog

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

Crossrefs

Cf. A206848 (exp), A226234, A206847, A206850, A206851.

Sequence in context: A274695 A156515 A254223 * A368395 A090907 A159478

Adjacent sequences: A206846 A206847 A206848 * A206850 A206851 A206852

Keyword

nonn

Author

Paul D. Hanna, Feb 15 2012

Status

approved