A129261 a(n) = coefficient of x^n in Product_{k=0..n} (1 + 2x + 3x^2 +...+ (k+1)*x^k).

1, 2, 7, 36, 204, 1222, 7513, 46950, 296691, 1890232, 12118424, 78080402, 505134625, 3279051382, 21347213562, 139319046744, 911204289149, 5970941722698, 39192011365250, 257632856738690, 1695850232984011

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:
[1,4,18,94,528,3106,18798,115964, ...] = A006013 * A129261;
[1,6,33,188,1105,6660,40888,254510,...]= A006013^2 * A129261.

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

Cf. A006013.

Sequence in context: A302895 A302905 A249691 * A371545 A370471 A177258

Adjacent sequences: A129258 A129259 A129260 * A129262 A129263 A129264

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2007

Status

approved