|
%I #13 Feb 10 2015 10:42:39
%S 1,2,13,144,2273,46710,1184153,35733376,1251320145,49893169050,
%T 2232012515445,110722046632560,6032418472347265,358103844593876654,
%U 23007314730623658225,1590611390957425536000,117745011140615270168865
%N Central coefficient of Product_{k=0..n} (1+k*x)^2.
%H Vaclav Kotesovec, <a href="/A129256/b129256.txt">Table of n, a(n) for n = 0..354</a>
%F a(n) = (-1)^n*Sum_{k=0..n} Stirling1(n+1,k+1)*Stirling1(n+1,n-k+1). - _Paul D. Hanna_, Jul 16 2009
%F a(n) ~ c * d^n * (n-1)!, where d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.910814964568255..., c = 0.851946112888790982829578047527831525434714038256... . - _Vaclav Kotesovec_, Feb 10 2015
%e This sequence equals the central terms of the triangle in which the g.f. of row n is (1+x)^2*(1+2x)^2*(1+3x)^2*...*(1+n*x)^2, as illustrated by:
%e (1);
%e 1,(2),1;
%e 1,6,(13),12,4;
%e 1,12,58,(144),193,132,36;
%e 1,20,170,800,(2273),3980,4180,2400,576;
%e 1,30,395,3000,14523,(46710),100805,143700,129076,65760,14400; ...
%t Flatten[{1,Table[Coefficient[Expand[Product[(1+k*x),{k,0,n}]^2],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 10 2015 *)
%o (PARI) a(n)=polcoeff(prod(k=0,n,1+k*x)^2,n)
%o Contribution from _Paul D. Hanna_, Jul 16 2009: (Start)
%o (PARI) {Stirling1(n,k)=if(n==0,1,n!*polcoeff(binomial(x,n),k))}
%o {a(n)=(-1)^n*sum(k=0,n,Stirling1(n+1,k+1)*Stirling1(n+1,n-k+1))} (End)
%Y Cf. A008275 (Stirling1 numbers), A238261.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 06 2007
|
