A129179 -id:A129179 - cp's OEIS frontend

A201075 Irregular triangle read by rows: number of Schroeder paths of length n and weighted area n^2-k.

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 4, 3, 3, 3, 1, 1, 1, 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 10, 7, 6, 4, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 14, 17, 22, 25, 27, 31, 34, 34, 33, 31, 28, 21, 14, 10, 5, 1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 13, 16, 21, 26, 31, 37, 45, 54

Offset: 0

N. J. A. Sloane, Nov 26 2011

0 <= k <= n^2.

			Triangle begins:
1
1 1
1 1 1 2 1
1 1 1 2 3 4 3 3 3 1
1 1 1 2 3 4 5 7 8 9 10 11 10 7 6 4 1
...

Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.

Mirror image of A129179.

Cf. A129176, A201076, A201079, A201080, A201159.

gf = Expand /@ FixedPoint[1 + x # (1 + q Normal@# /. {x :> q^2 x}) + O[x]^7 &, 0];
Flatten[Reverse[CoefficientList[#, q]] & /@ CoefficientList[gf, x]] (* Andrey Zabolotskiy, Jan 03 2024 *)

More terms from Andrey Zabolotskiy, Jan 03 2024

A326676 Triangular array: T(n,k) equals the number of n triangle stacks of large Schröder type with k down-triangles in the bottom row of the stack.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 3, 1, 0, 0, 1, 3, 4, 1, 0, 0, 1, 3, 6, 5, 1, 0, 0, 0, 4, 7, 10, 6, 1, 0, 0, 0, 3, 10, 14, 15, 7, 1, 0, 0, 0, 2, 11, 21, 25, 21, 8, 1, 0, 0, 0, 1, 10, 28, 40, 41, 28, 9, 1, 0, 0, 0, 1, 9, 31, 60, 71, 63, 36, 10, 1

Offset: 0

Peter Bala, Jul 17 2019

We define two types of plane triangles of unit area - up-triangles with vertices at the lattice points (x, y), (x+1, y+1) and (x+2, y) and down-triangles with vertices at the lattice points (x, y), (x-1, y+1) and (x+1, y+1).
To construct a triangle stack of large Schröder type we start with a horizontal row of k contiguous down-triangles forming the base row of the stack. Subsequent rows of the stack are formed by placing up-triangles on some, all or none of the down-triangles of the previous row. In the spaces between pairs of adjacent up-triangles further down-triangles may be placed. For an example, see the illustration in the Links section. There is an obvious bijective correspondence between triangle stacks of large Schröder type with a base of k down-triangles and large Schröder paths of semilength k. For another version of this array see A129179.
For triangle stacks of small Schröder type, where the base row consists of contiguous up-triangles, see A224704.

			Triangle begins
   n\k  0  1   2   3   4   5   6   7   8   9  10
   - - - - - - - - - - - - - - - - - - - - - - -
   0 |  1
   1 |  0  1
   2 |  0  1   1
   3 |  0  0   2   1
   4 |  0  0   1   3   1
   5 |  0  0   1   3   4   1
   6 |  0  0   1   3   6   5   1
   7 |  0  0   0   4   7  10   6   1
   8 |  0  0   0   3  10  14  15   7   1
   9 |  0  0   0   2  11  21  25  21   8  1
  10 |  0  0   0   1  10  28  40  41  28  9  1
...

P. Bala, Example of triangle stack of large Schröder type
P. Bala, Triangle Stacks of large Schröder type

Row sums A088352. Column sums A006318. Cf. A047998, A129179, A224704.

O.g.f. as a continued fraction: (q marks the area of the stack and b marks down-triangles in the base of the stack)
A(q,b) = 1/(1 - q*b - q^2*b/(1 - q^3*b - q^4*b/(1 - q^5*b - q^6*b/( (...) )))) = 1 + b*q + (b + b^2)*q^2 + (2*b^2 + b^3)*q^3 + (b^2 + 3*b^3 + b^4)*q^4 + ....
A(q,b) = 1/(1 - (q + q^2)*b/(1 - q^4*b/(1 - (q^3 + q^6)*b/(1 - q^8*b/(1 - (q^5 + q^10)*b/(1 - q^12*b/( (...) ))))))).
O.g.f. as a ratio of q-series: N(q,b)/D(q,b), where N(q,b) = Sum_{n >= 0} (-1)^n*q^(2*n^2+2*n)*b^n/( (Product_{k = 1..n} 1 - q^(2*k)) * (Product_{k = 1..n+1} 1 - q^(2*k-1)*b) ) and D(q,b) = Sum_{n >= 0} (-1)^n*q^(2*n^2)*b^n/( (Product_{k = 1..n} 1 - q^(2*k)) * (Product_{k = 1..n} 1 - q^(2*k-1)*b) ).

A129180 Total area below all Schroeder paths of semilength n.

Original entry on oeis.org

0, 1, 11, 85, 583, 3785, 23843, 147437, 900559, 5453457, 32816315, 196531781, 1172634391, 6976059865, 41401814099, 245230349021, 1450162049695, 8563622372129, 50510963880299, 297627067200821, 1752169739791591, 10307304302433513, 60592569330907523

Offset: 0

Emeric Deutsch, Apr 08 2007

nonn

A Schroeder path of semilength n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis.

			a(2) = 11 because the areas below the Schroeder paths HH, HUD, UDH, UDUD, UHD and UUDD are 0,1,1,2,3 and 4, respectively.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Lara Bossinger and Martina Lanini, Following Schubert varieties under Feigin's degeneration of the flag variety, arXiv:1802.04320 [math.RT], 2018.

Cf. A002315, A006319, A129179.

g:=(1+z)*(1-z-sqrt(1-6*z+z^2))^2/4/z/(1-6*z+z^2): gser:=series(g,z=0,30): seq(coeff(gser,z,n),n=0..24);

CoefficientList[Series[(1 + x)*(1 - x - Sqrt[1 - 6*x + x^2])^2/(4*x*(1 - 6*x + x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 03 2016 *)

a(n):=sum((((sqrt(2)+1)^(2*k+1)-(1-sqrt(2))^(2*k)*sqrt(2)+(1-sqrt(2))^(2*k))*sum(binomial(n+1-k,i+2)*binomial(n-k+i,i),i,0,n-k+1))/(n-k+1),k,0,n); /* Vladimir Kruchinin, Mar 02 2016 */

a(n) = Sum_{k=0..n^2} k * A129179(n,k).
G.f.: (1+z)*(1-z-sqrt(1-6*z+z^2))^2/(4*z*(1-6*z+z^2)) (obtained by computing (dG/dt)_{t=1} where G=G(t,z) is defined by G(t,z) = 1+z*G(t,z)+t*z*G(t,t^2*z)G(t,z); see A129179).
a(n) = Sum_{k=0..n} 2*A002315(k)*(Sum_{i=0..n-k+1} binomial(n+1-k,i+2)*binomial(n-k+i,i))/(n-k+1). - Vladimir Kruchinin, Mar 02 2016
a(n) ~ 1/2 * (1+sqrt(2))^(2*n+1). - Vaclav Kotesovec, Mar 03 2016
D-finite with recurrence -(n+1)*(2*n-5)*a(n) +3*(4*n+1)*(2*n-5)*a(n-1) +(-76*n^2+228*n-89)*a(n-2) +3*(2*n-1)*(4*n-13)*a(n-3) -(2*n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Jul 26 2022

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A201075 Irregular triangle read by rows: number of Schroeder paths of length n and weighted area n^2-k.

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A326676 Triangular array: T(n,k) equals the number of n triangle stacks of large Schröder type with k down-triangles in the bottom row of the stack.

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A129180 Total area below all Schroeder paths of semilength n.

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