0,2

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(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:

[1,4,18,94,528,3106,18798,115964, ...] = A006013 * A129261;

[1,6,33,188,1105,6660,40888,254510,...]= A006013^2 * A129261.

(PARI) {a(n)=polcoeff(prod(k=1, n+1, (1 - (k+1)*x^k + k*x^(k+1))/(1-x)^2), n)}

Cf. A006013.

Sequence in context: A101514 A111908 A060814 this_sequence A018997 A018954 A019030

Adjacent sequences: A129258 A129259 A129260 this_sequence A129262 A129263 A129264

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Apr 06 2007