A069269 -id:A069269 - cp's OEIS frontend

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

1, 1, 1, 1, 2, 4, 1, 3, 9, 22, 1, 4, 15, 52, 140, 1, 5, 22, 91, 340, 969, 1, 6, 30, 140, 612, 2394, 7084, 1, 7, 39, 200, 969, 4389, 17710, 53820, 1, 8, 49, 272, 1425, 7084, 32890, 135720, 420732, 1, 9, 60, 357, 1995, 10626, 53820, 254475, 1068012, 3362260

Offset: 0

Henry Bottomley, Mar 12 2002

For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).

Antidiagonals of convolution matrix of Table 1.5, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

			Rows start
  1;
  1,   1;
  1,   2,   4;
  1,   3,   9,  22;
  1,   4,  15,  52, 140;
etc.

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Columns include A000012, A000027, A055999.
Right-hand diagonals include A002293, A069271, A006632.
Cf. A130458 (row sums).

A069270 := proc(n,k)
        binomial(n+3*k,k)*(n-k+1)/(n+2*k+1) ;
end proc: # R. J. Mathar, Oct 11 2015

Table[Binomial[n + 3 k, k] (n - k + 1)/(n + 2 k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 27 2019 *)

T(n, k) = C(n+3k, k)*(n-k+1)/(n+2k+1).
For n >= k+3: T(n, k) = T(n-2, k+1)-T(n-3, k+1).
T(n, n) = T(n+2, n-1) = C(4n, n)/(3n+1).

A110616 A convolution triangle of numbers based on A001764.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 12, 7, 3, 1, 55, 30, 12, 4, 1, 273, 143, 55, 18, 5, 1, 1428, 728, 273, 88, 25, 6, 1, 7752, 3876, 1428, 455, 130, 33, 7, 1, 43263, 21318, 7752, 2448, 700, 182, 42, 8, 1, 246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1

Offset: 0

Philippe Deléham, Sep 14 2005, Jun 15 2007

Reflected version of A069269. - Vladeta Jovovic, Sep 27 2006
With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. - David Callan, Aug 27 2009
Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - Philippe Deléham, Jan 27 2014
Antidiagonals of convolution matrix of Table 1.4, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

			Triangle begins:
       1;
       1,      1;
       3,      2,     1;
      12,      7,     3,     1;
      55,     30,    12,     4,    1;
     273,    143,    55,    18,    5,    1;
    1428,    728,   273,    88,   25,    6,   1;
    7752,   3876,  1428,   455,  130,   33,   7,  1;
   43263,  21318,  7752,  2448,  700,  182,  42,  8, 1;
  246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1;
  ...
From _Peter Bala_, Feb 04 2025: (Start)
The transposed array factorizes as an infinite product of upper triangular arrays:
  / 1               \^T   /1             \^T /1             \^T / 1            \^T
  | 1    1           |   | 1   1          | | 0  1           |  | 0  1          |
  | 3    2   1       | = | 2   1   1      | | 0  1   1       |  | 0  0  1       | ...
  |12    7   3   1   |   | 5   2   1  1   | | 0  2   1  1    |  | 0  0  1  1    |
  |55   30  12   4  1|   |14   5   2  1  1| | 0  5   2  1  1 |  | 0  0  2  1  1 |
  |...               |   |...             | |...             |  |...            |
where T denotes transposition and [1, 1, 2, 5, 14,...] is the sequence of Catalan numbers A000108. (End)

Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 27, 29.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 21.
Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
Sheng-Liang Yang and L. J. Wang, Taylor expansions for the m-Catalan numbers, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.

Successive columns: A001764, A006013, A001764, A006629, A102893, A006630, A102594, A006631; row sums: A098746; see also A092276.

Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 28 2017 *)

T(n,k):=((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1); /* Vladimir Kruchinin, Nov 01 2011 */

T(n, k) = Sum_{j>=0} T(n-1, k-1+j)*A000108(j); T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n.
G.f.: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 + ... is the GF for A001764. - David Callan, Aug 27 2009
T(n, k) = ((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1). - Vladimir Kruchinin, Nov 01 2011

A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.

Original entry on oeis.org

0, -3, 12, -36, 105, -303, 873, -2514, 7239, -20844, 60018, -172815, 497601, -1432785, 4125540, -11879019, 34204272, -98487276, 283582809, -816544155, 2351145189, -6769852758, 19493014119, -56127897168, 161613838746, -465348502119, 1339917609189, -3858138988821

Offset: 0

Roman Witula, Aug 27 2012

The Berndt-type sequence number 10 for the argument 2Pi/9 defined by the first trigonometric relation from the section "Formula" below. The sequence a(n) is connected with sequences A215917 and A215885 - see the respective formula.

We have A035045(n)=abs(a(n+1)/3) for every n=0,1,...,5 and A035045(7) + a(7)/3 = 1, A035045(8) - a(8)/3 = 10, A035045(9) + a(9)/3 = 63, and A035045(10) - a(10)/3 = 320 - all these four results-numbers are in A069269.

			We have a(2)=-4*a(1), a(3)=-3*a(2), a(6)/a(3) = -24.25, and a(9) = 579*a(3).

D. Chmiela and R. Witula, Two parametric quasi-Fibonacci numbers of the ninth order, (submitted, 2012).
R. Witula, Ramanujan type formulas for arguments 2Pi/7 and 2Pi/9, Demonstratio Math. (in press, 2012).

Index entries for linear recurrences with constant coefficients, signature (-3,0,1).

Cf. A215917, A215885, A215664.

LinearRecurrence[{-3, 0, 1}, {0, -3, 12}, 50]

a(n) = c(1)*(-c(2))^(-n) + c(2)*(-c(4))^(-n) + c(4)*(-c(1))^(-n), where c(j) := 2*cos(2*Pi*j/9).
a(n) = A215917(n+1) + A215917(n) - 2*(-1)^n*A215885(n).
G.f.: -3*x*(1-x)/(1+3*x-x^3).

cp's OEIS Frontend

A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

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A110616 A convolution triangle of numbers based on A001764.

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A215919 a(n) = -3*a(n-1) + a(n-3), with a(0)=0, a(1)=-3, a(2)=12.

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