OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..310
FORMULA
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.
a(n) ~ c * (27/4)^n / sqrt(n), where c = 0.198914972420941446825144557191209003291459959208... . - Vaclav Kotesovec, Feb 11 2015
EXAMPLE
a(2) = [x^2] (1 + 2x)*(1 + 2x + 3x^2) = [x^2] (1 + 4x + 7x^2 + 6x^3) = 7.
a(3) = [x^3] (1 + 2x)*(1 + 2x + 3x^2)*(1 + 2x + 3x^2 + 4x^3)
= [x^3] (1 + 6x + 18x^2 + 36x^3 + 49x^4 + 46x^5 + 24x^6) = 36.
This sequence is a diagonal in the triangle of successive products:
(1);
1,(2);
1,4,(7),6;
1,6,18,(36),49,46,24;
1,8,33,94,(204),354,497,562,501,326,120;
1,10,52,188,528,(1222),2406,4102,6116,7996,9132,9014,7541,5116,2556,720; ...
Lower diagonals are convolutions with this sequence and A006013:
MATHEMATICA
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 *)
PROG
(PARI) {a(n)=polcoeff(prod(k=1, n+1, (1 - (k+1)*x^k + k*x^(k+1))/(1-x)^2), n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 06 2007
STATUS
approved