A121690 G.f.: A(x) = Sum_{k>=0} x^k * (1+x)^(k*(k+1)/2).

1, 1, 2, 4, 10, 27, 81, 262, 910, 3363, 13150, 54135, 233671, 1053911, 4951997, 24177536, 122381035, 640937746, 3466900453, 19337255086, 111057640382, 655892813805, 3978591077096, 24760700544301, 157941950878839

Offset

0,3

Comments

a(n) is the number of length n permutations that simultaneously avoid the bivincular patterns (123,{2},{}) and (132,{},{2}). - Christian Bean, Jun 03 2015

Links

Seiichi Manyama, Table of n, a(n) for n = 0..685

Christian Bean, A Claesson, H Ulfarsson, Simultaneous Avoidance of a Vincular and a Covincular Pattern of Length 3, arXiv preprint arXiv:1512.03226, 2015

Formula

a(n) = Sum_{k=0..n} C(k*(k+1)/2,n-k).

a(n) = A131338(n+1, n*(n+1)/2 + 1) for n>=0, where triangle A131338 starts with a '1' in row 0 and then for n>0 row n consists of n '1's followed by the partial sums of the prior row. - Paul D. Hanna, Aug 30 2007

From Paul D. Hanna, Apr 24 2010: (Start)

Let q = (1+x), then g.f. A(x) equals the continued fraction:

A(x) = 1/(1 - q*x/(1 - (q^2-q)*x/(1 - q^3*x/(1 - (q^4-q^2)*x/(1 - q^5*x/(1- (q^6-q^3)*x/(1 - q^7*x/(1 - (q^8-q^4)*x/(1 - ...)))))))))

due to an identity of a partial elliptic theta function.

(End)

G.f.: Sum_{n>=0} x^n * Product_{k=1..n} (1 - x*(1+x)^(2*k-2))/(1 - x*(1+x)^(2*k-1)). - Paul D. Hanna, Mar 21 2018

Mathematica

Table[Sum[Binomial[k*(k+1)/2, n-k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Jun 03 2015 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(k*(k+1)/2, n-k))
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=0, n, x^m*prod(k=1, m, (1 - x*(1+x)^(2*k-2))/(1 - x*(1+x)^(2*k-1) + x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 21 2018

Crossrefs

Cf. A121689, A131338.

Sequence in context: A291210 A340635 A099950 * A138356 A287693 A275428

Adjacent sequences: A121687 A121688 A121689 * A121691 A121692 A121693

Keyword

nonn

Author

Paul D. Hanna, Aug 15 2006

Status

approved