|
%I #9 Feb 11 2015 08:20:27
%S 1,2,7,36,204,1222,7513,46950,296691,1890232,12118424,78080402,
%T 505134625,3279051382,21347213562,139319046744,911204289149,
%U 5970941722698,39192011365250,257632856738690,1695850232984011
%N a(n) = coefficient of x^n in Product_{k=0..n} (1 + 2x + 3x^2 +...+ (k+1)*x^k).
%H Vaclav Kotesovec, <a href="/A129261/b129261.txt">Table of n, a(n) for n = 0..310</a>
%F a(n) = [x^n] Product_{k=1..n+1} { (1 - (k+1)*x^k + k*x^(k+1))/(1-x)^2 } for n>0, with a(0)=1.
%F a(n) ~ c * (27/4)^n / sqrt(n), where c = 0.198914972420941446825144557191209003291459959208... . - _Vaclav Kotesovec_, Feb 11 2015
%e a(2) = [x^2] (1 + 2x)*(1 + 2x + 3x^2) = [x^2] (1 + 4x + 7x^2 + 6x^3) = 7.
%e a(3) = [x^3] (1 + 2x)*(1 + 2x + 3x^2)*(1 + 2x + 3x^2 + 4x^3)
%e = [x^3] (1 + 6x + 18x^2 + 36x^3 + 49x^4 + 46x^5 + 24x^6) = 36.
%e This sequence is a diagonal in the triangle of successive products:
%e (1);
%e 1,(2);
%e 1,4,(7),6;
%e 1,6,18,(36),49,46,24;
%e 1,8,33,94,(204),354,497,562,501,326,120;
%e 1,10,52,188,528,(1222),2406,4102,6116,7996,9132,9014,7541,5116,2556,720; ...
%e Lower diagonals are convolutions with this sequence and A006013:
%e [1,4,18,94,528,3106,18798,115964, ...] = A006013 * A129261;
%e [1,6,33,188,1105,6660,40888,254510,...]= A006013^2 * A129261.
%t Flatten[{1,Table[Coefficient[Expand[Product[Sum[(j+1)*x^j,{j,0,k}],{k,0,n}]],x^n],{n,1,20}]}] (* _Vaclav Kotesovec_, Feb 10 2015 *)
%o (PARI) {a(n)=polcoeff(prod(k=1,n+1,(1 - (k+1)*x^k + k*x^(k+1))/(1-x)^2),n)}
%Y Cf. A006013.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 06 2007
|
