|
%I #10 Mar 03 2014 11:07:37
%S 1,2,10,407,56746,30771252,115106662819,1446405270234360,
%T 53819202633553797290,12313337704248075967333334,
%U 12373818231445938048765251252260,33156027144321617106970597265032233270,409476940913917468665022448013012674533441891
%N a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2,k^2).
%C Ignoring initial term a(0), equals the logarithmic derivative of A207139.
%F Limit n->infinity a(n)^(1/n^2) = 2. - _Vaclav Kotesovec_, Mar 03 2014
%e L.g.f.: L(x) = 2*x + 10*x^2/2 + 407*x^3/3 + 56746*x^4/4 + 30771252*x^5/5 +...
%e where exponentiation equals the g.f. of A207139:
%e exp(L(x)) = 1 + 2*x + 7*x^2 + 147*x^3 + 14481*x^4 + 6183605*x^5 +...
%e By definition, the initial terms begin: a(0) = 1;
%e a(1) = C(1,0)*C(1,0), + C(1,1)*C(1,1);
%e a(2) = C(2,0)*C(4,0), + C(2,1)*C(4,1), + C(2,2)*C(4,4);
%e 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);
%e 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); ...
%e which is evaluated as:
%e a(1) = 1*1 + 1*1 = 2;
%e a(2) = 1*1 + 2*4 + 1*1 = 10;
%e a(3) = 1*1 + 3*9 + 3*126 + 1*1 = 407;
%e a(4) = 1*1 + 4*16 + 6*1820 + 4*11440 + 1*1 = 56746;
%e a(5) = 1*1 + 5*25 + 10*12650 + 10*2042975 + 5*2042975 + 1*1 = 30771252;
%e a(6) = 1*1 + 6*36 + 15*58905 + 20*94143280 + 15*7307872110 + 6*600805296 + 1*1 = 115106662819; ...
%t Table[Sum[Binomial[n,k] * Binomial[n^2,k^2], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 03 2014 *)
%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)*binomial(n^2,k^2))}
%o for(n=0,16,print1(a(n),", "))
%Y Cf. A207139 (exp), A206851, A207136, A207138, A167009.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Feb 15 2012
|
