A207140 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2).

1, 2, 10, 407, 56746, 30771252, 115106662819, 1446405270234360, 53819202633553797290, 12313337704248075967333334, 12373818231445938048765251252260, 33156027144321617106970597265032233270, 409476940913917468665022448013012674533441891

Offset

0,2

Comments

Ignoring initial term a(0), equals the logarithmic derivative of A207139.

Links

Table of n, a(n) for n=0..12.

Formula

Limit n->infinity a(n)^(1/n^2) = 2. - Vaclav Kotesovec, Mar 03 2014

Example

L.g.f.: L(x) = 2*x + 10*x^2/2 + 407*x^3/3 + 56746*x^4/4 + 30771252*x^5/5 +...
where exponentiation equals the g.f. of A207139:
exp(L(x)) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
By definition, the initial terms begin: a(0) = 1;
a(1) = C(1,0)*C(1,0), + C(1,1)*C(1,1);
a(2) = C(2,0)*C(4,0), + C(2,1)*C(4,1), + C(2,2)*C(4,4);
a(3) = C(3,0)*C(9,0), + C(3,1)*C(9,1), + C(3,2)*C(9,4), + C(3,3)*C(9,9);
a(4) = C(4,0)*C(16,0), + C(4,1)*C(16,1), + C(4,2)*C(16,4), + C(4,3)*C(16,9), + C(4,4)*C(16,16); ...
which is evaluated as:
a(1) = 1*1 + 1*1 = 2;
a(2) = 1*1 + 2*4 + 1*1 = 10;
a(3) = 1*1 + 3*9 + 3*126 + 1*1 = 407;
a(4) = 1*1 + 4*16 + 6*1820 + 4*11440 + 1*1 = 56746;
a(5) = 1*1 + 5*25 + 10*12650 + 10*2042975 + 5*2042975 + 1*1 = 30771252;
a(6) = 1*1 + 6*36 + 15*58905 + 20*94143280 + 15*7307872110 + 6*600805296 + 1*1 = 115106662819; ...

Mathematica

Table[Sum[Binomial[n, k] * 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, k)*binomial(n^2, k^2))}
for(n=0, 16, print1(a(n), ", "))

Crossrefs

Cf. A207139 (exp), A206851, A207136, A207138, A167009.

Sequence in context: A206152 A261007 A013034 * A059723 A334286 A265627

Adjacent sequences: A207137 A207138 A207139 * A207141 A207142 A207143

Keyword

nonn

Author

Paul D. Hanna, Feb 15 2012

Status

approved