A207570 G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^(3*k-2) - 1).

1, 1, 4, 34, 410, 6455, 125251, 2888305, 77157780, 2342972405, 79701049425, 3002132647515, 124039845584382, 5577660227565634, 271162541308698623, 14172237715785139175, 792418822364402364530, 47198077739119663907870, 2983413619934353599892285

Offset

0,3

Comments

Compare g.f. to: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - 1), which is the g.f. of A179525.

Links

Michael De Vlieger, Table of n, a(n) for n = 0..200

Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Formula

a(n) ~ GAMMA(2/3) * 2^(2*n-1/3) * 3^(2*n+5/6) * n^(n-1/6) / (exp(n+Pi^2/72) * Pi^(2*n+7/6)). - Vaclav Kotesovec, May 06 2014

Example

G.f.: A(x) = 1 + x + 4*x^2 + 34*x^3 + 410*x^4 + 6455*x^5 + 125251*x^6 +...
such that, by definition,
A(x) = 1 + ((1+x)-1) + ((1+x)-1)*((1+x)^4-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1) + ((1+x)-1)*((1+x)^4-1)*((1+x)^7-1)*((1+x)^10-1) +...

Mathematica

Join[{1}, Rest[With[{nn=20}, CoefficientList[Series[Sum[Product[ (1+x)^(3k-2)-1, {k, n}], {n, nn}], {x, 0, nn}], x]]]] (* Harvey P. Dale, Aug 20 2012 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, prod(k=1, m, (1+x)^(3*k-2)-1) +x*O(x^n)), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A179525, A207556, A207569, A207571.

Sequence in context: A196692 A197065 A141007 * A158839 A236964 A145349

Adjacent sequences: A207567 A207568 A207569 * A207571 A207572 A207573

Keyword

nonn

Author

Paul D. Hanna, Feb 18 2012

Status

approved