

A326793


The number of small Schröder paths such that the area between the path and the xaxis contains n uptriangles.


2



1, 1, 2, 5, 12, 30, 75, 188, 472, 1186, 2981, 7494, 18842, 47376, 119126, 299545, 753220, 1894018, 4762640, 11976010, 30114592, 75725485, 190417684, 478820320, 1204031670, 3027633300, 7613224740, 19144059492, 48139261637, 121050006438
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OFFSET

0,3


COMMENTS

We define two types of plane triangles  uptriangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and downtriangles with vertices at the integer lattice points (x, y), (x1, y+1) and (x+1, y+1). The area beneath a small Schröder path may be decomposed in a unique manner into a collection of up and downtriangles. This decomposition produces a triangle stack in the sense of A224704. Here we are counting triangle stacks containing n uptriangles. See the Links section for an illustration.


LINKS

Table of n, a(n) for n=0..29.
P. Bala, Illustration for a(3) = 5


FORMULA

O.g.f. as a continued fraction: (u marks uptriangles)
A(u) = 1/(1  u/(1  u  u^2/(1  u^2  u^3/(1  u^3  u^4/(1  u^4  (...) ))))) = 1 + u + 2*u^2 + 5*u^3 + 12*u^4 + ....
A(u) = 1/(1  u/(1  (u + u^2)/(1  u^3/(1  (u^2 + u^4)/(1  u^5/(1  (u^3 + u^6)/(1  u^7/( (...) )))))))).
A(u) = 1/(2  (1 + u)/(2  (1 + u^2)/(2  (1 + u^3)/(2  (...) )))).
A(u) = N(u)/D(u), where N(u) = Sum_{n >= 0} u^(n^2+n)/ Product_{k = 1..n} ((1  u^k)^2) and D(u) = Sum_{n >= 0} u^(n^2)/ Product_{k = 1..n} ((1  u^k)^2).
a(n) ~ c*d^n, where c = 0.29475 98606 22204 98206 41002 ..., d = 2.51457 96438 78729 18851 04371 ....
Row sums of A326792.


CROSSREFS

Cf. A224704, A326792.
Sequence in context: A000106 A076883 A140832 * A026580 A092247 A331233
Adjacent sequences: A326790 A326791 A326792 * A326794 A326795 A326796


KEYWORD

nonn,easy


AUTHOR

Peter Bala, Jul 25 2019


STATUS

approved



