A103136 -id:A103136 - cp's OEIS frontend

A367393 a(n) = A103136(2*n, n), the central terms of the inverse of the Delannoy triangle.

1, -3, 22, -194, 1838, -18082, 182054, -1861890, 19258078, -200898626, 2109785654, -22275498434, 236225927182, -2514344180194, 26845804973638, -287403142763522, 3084015902579646, -33160937871888514, 357206218412519510, -3853959574555396290, 41640758821142160110

Offset: 0

Peter Luschny, Nov 16 2023

sign

A combinatorial interpretation in terms of Schroeder paths is given in A103136.

Cf. A103136.

# Using function A103136.
def A367393List(size):
    M = A103136(2*size)
    return [M[2*n][n] for n in range(size)]
print(A367393List(21))

A103137 First column of inverse of Delannoy triangle.

Original entry on oeis.org

1, -1, 2, -6, 22, -90, 394, -1806, 8558, -41586, 206098, -1037718, 5293446, -27297738, 142078746, -745387038, 3937603038, -20927156706, 111818026018, -600318853926, 3236724317174, -17518619320890, 95149655201962, -518431875418926, 2832923350929742, -15521467648875090

Offset: 0

Paul Barry, Jan 24 2005

First column of A103136. The positive sequence has g.f. 1+xS(x). It is the first column of the inverse of the signed Delannoy triangle which has general term T(n,k)=if(k<=n, sum{j=0..k, 2^j*C(n-k,j)C(k,j)}(-1)^(n-k),0).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

A minor variation of A006318.

CoefficientList[Series[1+x*(1+x-(1+6*x+x^2)^(1/2))/(2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

G.f.: 1-xS(-x), where S(x) is the g.f. of the large Schroeder numbers A006318.
Conjecture: n*a(n) +3*(2*n-3)*a(n-1) +(n-3)*a(n-2)=0. - R. J. Mathar, Nov 26 2012
a(n) ~ (-1)^n * sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 13 2014

A132372 T(n, k) counts Schroeder n-paths whose ascent starting at the initial vertex has length k. Triangle T(n,k), read by rows.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 10, 5, 1, 22, 38, 22, 7, 1, 90, 158, 98, 38, 9, 1, 394, 698, 450, 194, 58, 11, 1, 1806, 3218, 2126, 978, 334, 82, 13, 1, 8558, 15310, 10286, 4942, 1838, 526, 110, 15, 1, 41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1

Offset: 0

Philippe Deléham, Nov 20 2007

Triangle T(n,k), 0<=k<=n, read by rows given by [1,1,2,1,2,1,2,1,2,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .
Transpose of triangular array A033878. - Michel Marcus, May 02 2015
The triangle is the Riordan square (A321620) of A155069. - Peter Luschny, Feb 01 2020

			Triangle begins:
      1;
      1,     1;
      2,     3,     1;
      6,    10,     5,     1;
     22,    38,    22,     7,    1;
     90,   158,    98,    38,    9,    1;
    394,   698,   450,   194,   58,   11,   1;
   1806,  3218,  2126,   978,  334,   82,  13,   1;
   8558, 15310, 10286,  4942, 1838,  526, 110,  15,  1;
  41586, 74614, 50746, 25150, 9922, 3142, 778, 142, 17, 1 ; ...
...
The production matrix M begins:
  1, 1
  1, 2, 1
  1, 2, 2, 1
  1, 2, 2, 2, 1
  1, 2, 2, 2, 2, 1
  ...

Lili Mu and Sai-nan Zheng, On the Total Positivity of Delannoy-Like Triangles, Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.6.
E. Pergola and R. A. Sulanke, Schroeder Triangles, Paths and Parallelogram Polyominoes, Journal of Integer Sequences, Vol. 1 (1998), #98.1.7.

Cf. A006318, A103136 (signed version), A033878 (transpose).

Cf. A155069, A321620.

# The function RiordanSquare is defined in A321620.
RiordanSquare((3-x-sqrt(1-6*x+x^2))/2, 10); # Peter Luschny, Feb 01 2020
# Alternative:
A132372 := proc(dim) # dim is the number of rows requested.
local T, j, A, k, C, m; m := 1;
T := [seq([seq(0, j = 0..k)], k = 0..dim-1)];
A := [seq(ifelse(k = 0, 1 + x, 2 - irem(k, 2)), k = 0..dim-2)];
C := [seq(1, k = 1..dim+1)]; C[1] := 0;
for k from 0 to dim - 1 do
    for j from k + 1 by -1 to 2 do
        C[j] := C[j-1] + C[j+1] * A[j-1] od;
    T[m] := [seq(coeff(C[2], x, j), j = 0..k)];
    m := m + 1
od; ListTools:-Flatten(T) end:
A132372(10);  # Peter Luschny, Nov 16 2023

Sum_{k, 0<=k<=n} T(n,k) = A006318(n) .

T(n,0) = A155069(n). - Philippe Deléham, Nov 03 2009

A103138 Second column of inverse of Delannoy triangle.

Original entry on oeis.org

0, 1, -3, 10, -38, 158, -698, 3218, -15310, 74614, -370610, 1869338, -9549174, 49302030, -256859754, 1348695330, -7129819038, 37916710374, -202708895330, 1088819681834, -5873129780422, 31800514324606, -172780691083034, 941714095635890, -5147414826440558, 28210011946820438

Offset: 0

Paul Barry, Jan 24 2005

The positive sequence has g.f. (1+x*S(x))*x*S(x).

Second column of A103136.

			G.f.: A(x) = x - 3*x^2 + 10*x^3 - 38*x^4 + 158*x^5 - 698*x^6 + ... where A( x*(1+x)/(1-x) ) / (1-x) = x.

Vincenzo Librandi, Table of n, a(n) for n = 0..200

CoefficientList[Series[(1-x*(1+x-(1+6*x+x^2)^(1/2))/(-2*x))*x*(1+x-(1+6*x+x^2)^(1/2))/(-2*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 01 2014 *)

{a(n)=if(n==1, 1, -polcoeff(sum(k=1, n-1, a(k)*x^k*(1+x)^k/(1-x+x*O(x^n))^(k+1)), n))} \\ Paul D. Hanna, Aug 06 2013

G.f.: (1-x*S(-x))*x*S(-x), where S(x) is the g.f. of the large Schroeder numbers A006318.
Conjecture: 2*n*a(n) +(13*n-20)*a(n-1) +(8*n-27)*a(n-2) +(n-5)*a(n-3)=0. - R. J. Mathar, Dec 14 2011
G.f.: x = Sum_{n>=1} a(n) * x^n * (1+x)^n / (1-x)^(n+1). - Paul D. Hanna, Aug 06 2013
G.f. satisfies: A(x*(1+x)/(1-x)) = x - x^2. - Paul D. Hanna, Aug 06 2013
a(n) ~ (-1)^n * (1-2*sqrt(2)) * sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 01 2014

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A367393 a(n) = A103136(2*n, n), the central terms of the inverse of the Delannoy triangle.

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A103137 First column of inverse of Delannoy triangle.

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A132372 T(n, k) counts Schroeder n-paths whose ascent starting at the initial vertex has length k. Triangle T(n,k), read by rows.

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A103138 Second column of inverse of Delannoy triangle.

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