A047110 -id:A047110 - cp's OEIS frontend

A047130 Array read by descending antidiagonals: T(h,k) is the number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no up-step crosses the line y = 3x/4. (Thus a path crosses the line only at lattice points and on right-steps.)

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 5, 4, 1, 1, 5, 9, 5, 9, 5, 1, 1, 6, 14, 14, 14, 14, 6, 1, 1, 7, 20, 28, 28, 28, 20, 7, 1, 1, 8, 27, 48, 56, 28, 48, 27, 8, 1, 1, 9, 35, 75, 104, 84, 76, 75, 35, 9, 1, 1, 10, 44, 110, 179, 188, 84

Offset: 0

Clark Kimberling. Definition revised Dec 08 2006

			Array begins:
===================================
h\k | 0 1  2  3   4   5   6   7
----+------------------------------
  0 | 1 1  1  1   1   1   1   1 ...
  1 | 1 1  2  3   4   5   6   7 ...
  2 | 1 2  2  5   9  14  20  27 ...
  3 | 1 3  5  5  14  28  48  75 ...
  4 | 1 4  9 14  28  56 104 179 ...
  5 | 1 5 14 28  28  84 188 367 ...
  6 | 1 6 20 48  76  84 272 639 ...
  7 | 1 7 27 75 151 235 272 911 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Cf. A047131, A047132, A047133, A047134, A047135, A047136, A047137, A047138, A047139.

Cf. A047110, A047140, A047150.

A(h, k=h)={my(M=matrix(h+1, k+1, i, j, 1)); for(h=1, h, for(k=1, k, M[1+h, 1+k] = M[h, 1+k] + if(4*k>3*h && 4*(k-1)<3*h, 0, M[1+h, k]))); M}
{ my(T=A(10)); for(i=1, #T, print(T[i, ]))} \\ Andrew Howroyd, Jan 19 2020

A047140 Array read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no up-step crosses the line y=4x/3. (Thus a path crosses the line only at lattice points and on right-steps.).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 4, 4, 1, 1, 3, 2, 8, 5, 1, 1, 4, 5, 10, 13, 6, 1, 1, 5, 9, 15, 23, 19, 7, 1, 1, 6, 14, 24, 38, 42, 26, 8, 1, 1, 7, 20, 38, 62, 80, 68, 34, 9, 1, 1, 8, 27, 58, 38, 142, 148, 102, 43, 10, 1, 1, 9, 35, 85, 96, 180, 290

Offset: 0

Clark Kimberling. Definition revised Dec 08 2006

			Array begins:
======================================
h\k | 0 1  2   3   4   5    6    7
----+---------------------------------
  0 | 1 1  1   1   1   1    1    1 ...
  1 | 1 2  1   2   3   4    5    6 ...
  2 | 1 3  4   2   5   9   14   20 ...
  3 | 1 4  8  10  15  24   38   58 ...
  4 | 1 5 13  23  38  62   38   96 ...
  5 | 1 6 19  42  80 142  180   96 ...
  6 | 1 7 26  68 148 290  470  566 ...
  7 | 1 8 34 102 250 540 1010 1576 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Cf. A047141, A047142, A047143, A047144, A047145, A047146, A047147, A047148, A047149.

Cf. A047110, A047130, A047150.

A(h,k=h)={my(M=matrix(h+1,k+1,i,j,1)); for(h=1, h, for(k=1, k, M[1+h, 1+k] = M[h,1+k] + if(3*k>4*h && 3*(k-1)<4*h, 0, M[1+h,k]))); M}
{ my(T=A(10)); for(i=1, #T, print(T[i,]))} \\ Andrew Howroyd, Jan 19 2020

A047099 a(n) = A047098(n)/2.

Original entry on oeis.org

1, 4, 19, 98, 531, 2974, 17060, 99658, 590563, 3540464, 21430267, 130771376, 803538100, 4967127736, 30866224824, 192696614730, 1207967820099, 7600482116932, 47981452358201, 303820299643138, 1929099000980219, 12279621792772822, 78346444891033856

Offset: 1

Clark Kimberling, Dec 11 1999

nonn

T(2*n,n)/2, with array T as in A047110.
Also given by a recurrence that features row 3 of the Pascal triangle (Mathematica code): u[0,0]=1; u[n_,k_]/;k<0 || k>n := 0; u[n_,k_]/;0<=k<=n := u[n,k] = u[n-1,k-1] + 3u[n-1,k] + 3u[n-1,k+1] + u[n-1,k+2]; u[n_]:=Sum[u[n,k],{k,0,n}]; Table[u[n],{n,0,10}]. - David Callan, Jul 22 2008
INVERT transform of (1,3,12,55,273,...), the ternary numbers A001764. - David Callan, Nov 21 2011

Alois P. Heinz, Table of n, a(n) for n = 1..400
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
J.-P. Bultel and S. Giraudo, Combinatorial Hopf algebras from PROs, arXiv preprint arXiv:1406.6903 [math.CO], 2014-2016.
Isaac DeJager, Madeleine Naquin, and Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.
Noga Alon and Noah Kravitz, Counting Dope Matrices, arXiv:2205.09302 [math.CO], 2022.

Cf. A066380, A005809.
Column k=2 of A213027.
Cf. A001764.

f := n -> binomial(3*n, n) - (1/2)*add(binomial(3*n, k), k=0..n):
seq(f(n), n=1..20);

Table[Binomial[3 n, n] - Sum[Binomial[3 n, k], {k, 0, n}]/2, {n, 20}] (* Wesley Ivan Hurt, Jun 13 2014 *)

{a(n)=local(A=1+x); A=x*exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, j))*x^m/m +x*O(x^n))); polcoeff(A, n)} \\ Paul D. Hanna, Sep 04 2012

a(n) = binomial(3*n, n) - (1/2)*Sum_{k=0..n} binomial(3*n, k). - Vladeta Jovovic, Mar 22 2003
a(n) = A047098(n)/2. - Benoit Cloitre, Jan 28 2004
From Gary W. Adamson, Jul 28 2011: (Start)
a(n) is the upper left term in M^n, where M is the infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  3, 3, 1, 0, 0, 0, ...
  3, 3, 3, 1, 0, 0, ...
  1, 1, 3, 3, 1, 0, ...
  0, 0, 1, 3, 3, 0, ...
  0, 0, 0, 1, 3, 0, ...
  ... (End)
G.f.: x*exp( Sum_{n>=1} A066380*x^n/n ) where A066380(n) = Sum_{k=0..n} binomial(3*n,k). - Paul D. Hanna, Sep 04 2012
G.f.: (F(x)-1)/(2-F(x)), where F(x) is g.f. of A001764. - Vladimir Kruchinin, Jun 13 2014.
a(n) = (1/n)*Sum_{k=1..n} k*C(3*n,n-k). - Vladimir Kruchinin, Oct 03 2022
From Paul D. Hanna, Jun 06 2025: (Start)
G.f. A(x) = Series_Reversion( x*(1 + x)^2 / (1 + 2*x)^3 ).
G.f. satisfies A(x) = x*(1 + 2*A(x))^3 / (1 + A(x))^2.
G.f. satisfies A'(x) = A(x) * (1 + A(x)) * (1 + 2*A(x)) / (x*(1 - A(x))).
(End)

Comment revised by Clark Kimberling, Dec 08 2006

Edited by N. J. A. Sloane, Dec 21 2006

A125778 Array T by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no right-step crosses the line y=3x/2. (Thus a path crosses the line only at lattice points and on up-steps.).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 2, 3, 1, 1, 4, 5, 5, 4, 1, 1, 5, 9, 10, 9, 5, 1, 1, 6, 14, 10, 19, 14, 6, 1, 1, 7, 20, 24, 29, 33, 20, 7, 1, 1, 8, 27, 44, 29, 62, 53, 27, 8, 1, 1, 9, 35, 71, 73, 91, 115, 80, 35, 9, 1, 1, 10, 44, 106, 144, 164, 206, 195, 115, 44, 10, 1, 1, 11, 54, 150

Offset: 0

Clark Kimberling, Dec 08 2006

T = transpose of A047110.

			Northwest corner:
1 1 1 1 1 1
1 1 2 3 4 5
1 2 2 5 9 14
1 3 5 10 10 24
1 4 9 19 29 29

Cf. A047110.

A127832 Array T by antidiagonals: for h>=0 and k>=0, T(h,k)=number of UR paths from (0,0) to (h,k) that touch the line y=2x/3 only at lattice points. A UR path is a path of steps of length 1 each directed up or right.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 2, 2, 1, 1, 4, 5, 4, 3, 1, 1, 5, 9, 9, 7, 4, 1, 1, 6, 14, 18, 9, 11, 5, 1, 1, 7, 20, 32, 27, 11, 16, 6, 1, 1, 8, 27, 52, 59, 27, 27, 22, 7, 1, 1, 9, 35, 79, 111, 86, 54, 49, 29, 8, 1, 1, 10, 44, 114, 190, 197, 140, 103, 78, 37, 9, 1, 1, 11, 54, 158, 304, 387

Offset: 1

Clark Kimberling, Feb 01 2007

			Northwest corner:
0 1 1 1 1 1
1 1 2 3 4 5
1 1 2 5 9 14
1 2 4 9 18 32
1 3 7 9 27 59
The 4 UR paths for T(3,2) are RRRUU, RRURU, UURRR, URURR. These
paths touch the line y=2x/3 only at the lattice points (0,0) and (3,2).

Cf. A047110, A127833.

cp's OEIS Frontend

A047130 Array read by descending antidiagonals: T(h,k) is the number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no up-step crosses the line y = 3x/4. (Thus a path crosses the line only at lattice points and on right-steps.)

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A047140 Array read by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no up-step crosses the line y=4x/3. (Thus a path crosses the line only at lattice points and on right-steps.).

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A047099 a(n) = A047098(n)/2.

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A125778 Array T by diagonals: T(h,k)=number of paths consisting of steps from (0,0) to (h,k) such that each step has length 1 directed up or right and no right-step crosses the line y=3x/2. (Thus a path crosses the line only at lattice points and on up-steps.).

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A127832 Array T by antidiagonals: for h>=0 and k>=0, T(h,k)=number of UR paths from (0,0) to (h,k) that touch the line y=2x/3 only at lattice points. A UR path is a path of steps of length 1 each directed up or right.

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