A061554 -id:A061554 - cp's OEIS frontend

A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

1, 1, 1, 5, 1, 1, 21, 6, 1, 1, 93, 25, 7, 1, 1, 421, 112, 29, 8, 1, 1, 1937, 510, 132, 33, 9, 1, 1, 9017, 2357, 606, 153, 37, 10, 1, 1, 42349, 11009, 2819, 709, 175, 41, 11, 1, 1, 200277, 51840, 13233, 3324, 819, 198, 45, 12, 1, 1

Offset: 0

Philippe Deléham, Dec 06 2009

Expansion of row sums of T_(x,3), T_(x,y) defined in A039599.

Matrix product P^3 * Q * P^(-3), where P denotes Pascal's triangle A007318 and Q denotes A061554 (formed from P by sorting the rows into descending order). Cf. A158793 and A158815. - Peter Bala, Jul 13 2021

			Triangle begins:
    1;
    1,   1;
    5,   1,  1;
   21,   6,  1, 1;
   93,  25,  7, 1, 1;
  421, 112, 29, 8, 1, 1;
  ...

Cf. A061554, A158793, A158815, A171224.

Sum_{k=0..n} T(n,k)*x^k = A126952(n), A126568(n), A026375(n), A026378(n+1), A000351(n) for x = 0,1,2,3,4 respectively.

A126125 Triangle equal to the matrix square of the triangle binomial(n,floor((n+1-(-1)^(n+k)*(k+1))/2)).

Original entry on oeis.org

1, 2, 1, 5, 2, 1, 11, 7, 2, 1, 27, 15, 9, 2, 1, 61, 44, 19, 11, 2, 1, 149, 97, 65, 23, 13, 2, 1, 342, 267, 141, 90, 27, 15, 2, 1, 835, 599, 433, 193, 119, 31, 17, 2, 1, 1939, 1598, 956, 655, 253, 152, 35, 19, 2, 1, 4740, 3631

Offset: 0

Gary W. Adamson, Dec 17 2006

The triangular view of A061554 is:
1;
1, 1;
2, 1, 1;
3, 3, 1, 1;
6, 4, 4, 1, 1;
The matrix square of this (infinite) triangle is this triangle here.

			First few rows of the triangle are:
1;
2, 1;
5, 2, 1;
11, 7, 2, 1;
27, 15, 9, 2, 1;
61, 44, 19, 11, 2, 1;
..,

Cf. A061554, A127358 (row sums).

A061554 := proc(n,k) local m ; m := floor((n+1)/2 - (-1)^(n-k)*(k+1)/2) ; binomial(n,m) ; end proc:
A126125 := proc(n,k) add(A061554(n,j)*A061554(j,k),j=k..n) ; end proc: # R. J. Mathar, Sep 17 2011

A298637 Triangular array of a Catalan number variety: T(n,k) is the number of words consisting of n parentheses containing k well-balanced pairs.

Original entry on oeis.org

1, 2, 3, 1, 4, 4, 5, 9, 2, 6, 16, 10, 7, 25, 27, 5, 8, 36, 56, 28, 9, 49, 100, 84, 14, 10, 64, 162, 192, 84, 11, 81, 245, 375, 270, 42, 12, 100, 352, 660, 660, 264, 13, 121, 486, 1078, 1375, 891, 132, 14, 144, 650, 1664, 2574, 2288, 858, 15, 169, 847, 2457, 4459, 5005, 3003, 429

Offset: 0

Marko Riedel, Jan 23 2018

A well-balanced run in a word of parentheses is a maximal run where every initial segment of the run has at least as many left parentheses as right ones and the number of open parentheses is the same as that of closed ones. The variable k in the sequence definition is the sum of the count of balanced pairs in all maximal runs in the word and n is the length of the word. Runs are maximal substrings counted by ordinary Catalan numbers.

			The word ))))(()(()))((() contains five well-balanced pairs of parentheses.
Triangular array T(n,k) begins:
   1;
   2;
   3,   1;
   4,   4;
   5,   9,   2;
   6,  16,  10;
   7,  25,  27,   5;
   8,  36,  56,  28;
   9,  49, 100,  84,  14;
  10,  64, 162, 192,  84;
  11,  81, 245, 375, 270,  42;
  12, 100, 352, 660, 660, 264;

Alois P. Heinz, Rows n = 0..200, flattened
Toufik Mansour, Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
Marko Riedel et al., Generalisation for Catalan number.
Marko Riedel, Maple code for A298637 including enumeration, generating function, and two closed forms.

Row sums give A000079.
T(2n,n) gives A000108.
T(2n+1,n) gives A068875. T(n,1) gives A000290. T(2n,2) gives A280089.
Cf. A007318, A061554.

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
     `if`(i>0, x, 1)*b(n-1, max(0, i-1))+b(n-1, i+1)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..16);  # Alois P. Heinz, Jan 23 2018

Table[((n + 1 - 2 k)^2/(n + 1)) Binomial[n + 1, k], {n, 0, 17}, {k, 0, Floor[n/2]}] // Flatten (* Michael De Vlieger, Jan 23 2018 *)

T(n,k) = ((n+1-2*k)^2/(n+1))*C(n+1,k) where 0 <= k <= floor(n/2).
Bivariate o.g.f. is C(u*z^2)/(1-z*C(u*z^2))^2 with u counting pairs of parentheses and z counting total word length where C(z) = (1-sqrt(1-4*z))/(2*z) is the o.g.f. of the Catalan numbers.
T(2*k,k) = C(k), the k-th Catalan number.
T(n,0) = n+1 by construction.

A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 4, 4, 1, 1, 10, 5, 5, 1, 1, 15, 15, 6, 6, 1, 1, 35, 21, 21, 7, 7, 1, 1, 56, 56, 28, 28, 8, 8, 1, 1, 126, 84, 84, 36, 36, 9, 9, 1, 1, 210, 210, 120, 120, 45, 45, 10, 10, 1, 1, 462, 330, 330, 165, 165, 55, 55, 11, 11, 1, 1, 792, 792, 495, 495, 220, 220, 66, 66, 12, 12, 1, 1

Offset: 1

Stefano Spezia, May 31 2020

T(n, k) is a tight upper bound of the cardinality of an intersecting Sperner family or antichain of the set {1, 2,..., n}, where every collection of pairwise independent subsets is characterized by an intersection of cardinality at least k (see Theorem 1.3 in Wong and Tay).

Equals A061554 with the first row of the array (resp. the first column of the triangle) removed. - Georg Fischer, Jul 26 2023

			The triangle T(n, k) begins
n\k|  1   2   3   4   5   6   7   8
---+-------------------------------
1  |  1
2  |  1   1
3  |  3   1   1
4  |  4   4   1   1
5  | 10   5   5   1   1
6  | 15  15   6   6   1   1
7  | 35  21  21   7   7   1   1
8  | 56  56  28  28   8   8   1   1
...

Eric Charles Milner, A Combinatorial Theorem On Systems of Sets, Journal of the London Mathematical Society, 43, (1968), 204-206.
W. H. W. Wong and E. G. Tay, On Cross-intersecting Sperner Families, arXiv:2001.01910 [math.CO], 2020.
Index entries for sequences related to antichains.

Cf. A000372, A001405, A004526, A007318, A007695, A061554, A266696, A325982, A325983.

Cf. A037951 (k=3), A037952 (k=1), A037953 (k=5), A037954 (k=7), A037955 (k=2), A037956 (k=4), A037957 (k=6), A037958 (k=8), A045621 (row sums).

T[n_,k_]:=Binomial[n,Floor[(n+k+1)/2]]; Table[T[n,k],{n,12},{k,n}]//Flatten

T(n, k) = binomial(n, (n+k+1)\2);
vector(10, n, vector(n, k, T(n, k))) \\ Michel Marcus, Jun 01 2020

T(n, k) = A007318(n, A004526(n+k+1)) with k <= n.

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A171243 Riordan array (f(x), x*g(x)), f(x) is the g.f. of A126952, g(x) is the g.f. of A117641.

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A126125 Triangle equal to the matrix square of the triangle binomial(n,floor((n+1-(-1)^(n+k)*(k+1))/2)).

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A298637 Triangular array of a Catalan number variety: T(n,k) is the number of words consisting of n parentheses containing k well-balanced pairs.

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A335322 Triangle read by rows: T(n, k) = binomial(n, floor((n+k+1)/2)) with k <= n.

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