reviewed
approved
Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
reviewed
approved
proposed
reviewed
editing
proposed
We define two types of plane triangles - up-triangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the integer lattice points (x, y), (x-1, 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 down-triangles. This decomposition produces a triangle stack in the sense of A224704. Here we are counting triangle stacks containing n up-triangles. See the Links section for an illustration.
proposed
editing
editing
proposed
a(n) ~ c*d^n, where c = 0.29475 98606 22204 98206 41002 ..., d = 2.51457 96438 78729 18851 043719 04371 ....
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(n) ~ c*d^n, where c = 0.29475 98606 22204 98206 41002 ..., d = 2.51457 96438 78729 18851 043719 ....
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 + .... Row sums of A326792.
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 043719 ....
Row sums of A326792.
allocated for Peter BalaThe number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles.
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
0,3
We define two types of plane triangles - up-triangles with vertices at the integer lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the integer lattice points (x, y), (x-1, 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 down-triangles. This decomposition produces a triangle stack in the sense of A224704. Here we are counting triangle stacks containing n up-triangles. See the Links section for an illustration.
P. Bala, <a href="/A326793/a326793.pdf">Illustration for a(3) = 5</a>
O.g.f. as a continued fraction: (u marks up-triangles)
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 + .... Row sums of A326792.
allocated
nonn,easy
Peter Bala, Jul 25 2019
approved
editing
allocated for Peter Bala
allocated
approved