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Revision History for A326793 (Bold, blue-underlined text is an addition; faded, red-underlined text is a deletion.)

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The number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles.
(history; published version)
#10 by Peter Luschny at Mon Jul 29 12:00:28 EDT 2019
STATUS

reviewed

approved

#9 by Joerg Arndt at Mon Jul 29 10:19:41 EDT 2019
STATUS

proposed

reviewed

#8 by Jon E. Schoenfield at Sun Jul 28 20:11:46 EDT 2019
STATUS

editing

proposed

#7 by Jon E. Schoenfield at Sun Jul 28 20:11:44 EDT 2019
COMMENTS

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.

STATUS

proposed

editing

#6 by Peter Bala at Sun Jul 28 14:08:56 EDT 2019
STATUS

editing

proposed

#5 by Peter Bala at Thu Jul 25 18:35:44 EDT 2019
FORMULA

a(n) ~ c*d^n, where c = 0.29475 98606 22204 98206 41002 ..., d = 2.51457 96438 78729 18851 043719 04371 ....

#4 by Peter Bala at Thu Jul 25 18:34:33 EDT 2019
FORMULA

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 ....

#3 by Peter Bala at Thu Jul 25 18:33:13 EDT 2019
FORMULA

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.

#2 by Peter Bala at Thu Jul 25 16:11:06 EDT 2019
NAME

allocated for Peter BalaThe number of small Schröder paths such that the area between the path and the x-axis contains n up-triangles.

DATA

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

OFFSET

0,3

COMMENTS

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.

LINKS

P. Bala, <a href="/A326793/a326793.pdf">Illustration for a(3) = 5</a>

FORMULA

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.

CROSSREFS
KEYWORD

allocated

nonn,easy

AUTHOR

Peter Bala, Jul 25 2019

STATUS

approved

editing

#1 by Peter Bala at Thu Jul 25 15:26:56 EDT 2019
NAME

allocated for Peter Bala

KEYWORD

allocated

STATUS

approved