A001453 -id:A001453 - cp's OEIS frontend

A289616 A246604 (Catalan(n)-n) with initial terms 1,0,0,2,10 changed to 1,1,1,2,11.

1, 1, 1, 2, 11, 37, 126, 422, 1422, 4853, 16786, 58775, 208000, 742887, 2674426, 9694830, 35357654, 129644773, 477638682, 1767263171, 6564120400, 24466266999, 91482563618, 343059613627, 1289904147300, 4861946401427, 18367353072126, 69533550915977, 263747951750332, 1002242216651339

Offset: 1

N. J. A. Sloane, Jul 09 2017

nonn

Related to number of mesh patterns of length 2 that avoid the pattern 321.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

A variant of A246604.

All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.

Join[{1,1,1,2,11},Array[CatalanNumber[#]-#&,30,5]] (* Paolo Xausa, Dec 08 2023 *)

A289652 Catalan numbers - 2 (A120304) with first three terms changed to 1,1,1.

Original entry on oeis.org

1, 1, 1, 3, 12, 40, 130, 427, 1428, 4860, 16794, 58784, 208010, 742898, 2674438, 9694843, 35357668, 129644788, 477638698, 1767263188, 6564120418, 24466267018, 91482563638, 343059613648, 1289904147322, 4861946401450, 18367353072150, 69533550916002, 263747951750358

Offset: 1

N. J. A. Sloane, Jul 09 2017

nonn

Related to number of mesh patterns of length 2 that avoid the pattern 321.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

A variant of A120304.

All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.

Join[{1,1,1},CatalanNumber[Range[3,30]]-2] (* Paolo Xausa, Dec 08 2023 *)

A289653 Catalan numbers - 2 (A120304) with first four terms changed to 1,1,1,4.

Original entry on oeis.org

1, 1, 1, 4, 12, 40, 130, 427, 1428, 4860, 16794, 58784, 208010, 742898, 2674438, 9694843, 35357668, 129644788, 477638698, 1767263188, 6564120418, 24466267018, 91482563638, 343059613648, 1289904147322, 4861946401450, 18367353072150, 69533550916002, 263747951750358

Offset: 1

N. J. A. Sloane, Jul 09 2017

nonn

Related to number of mesh patterns of length 2 that avoid the pattern 321.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

A variant of A120304.All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.

Join[{1,1,1,4},CatalanNumber[Range[4,30]]-2] (* Paolo Xausa, Dec 08 2023 *)

A289654 Related to number of mesh patterns of length 2 that avoid the pattern 321.

Original entry on oeis.org

1, 1, 1, 3, 13, 40, 130, 427, 1428, 4860, 16794

Offset: 1

N. J. A. Sloane, Jul 09 2017

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

All of A000108, A001453, A246604, A273526, A120304, A289615, A289616, A289652, A289653, A289654 are very similar sequences.

A236855 a(n) is the sum of digits in A239903(n).

Original entry on oeis.org

0, 1, 1, 2, 3, 1, 2, 2, 3, 4, 3, 4, 5, 6, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 3, 4, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 10, 1, 2, 2, 3, 4, 2, 3, 3, 4, 5, 4, 5, 6, 7, 2, 3, 3, 4, 5, 3, 4, 4, 5, 6, 5, 6, 7, 8, 4, 5, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 11, 3, 4

Offset: 0

Antti Karttunen, Apr 18 2014

			As the 0th Catalan String is empty, indicated by A239903(0)=0, a(0)=0.
As the 18th Catalan String is [1,0,1,2] (A239903(18)=1012), a(18) = 1+0+1+2 = 4.
Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 55, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.

Antti Karttunen, Table of n, a(n) for n = 0..16796

Cf. A239903, A014420, A237449, A236859, A009766, A071155, A081291, A034968, A060130.

A236855list[m_] := With[{r = 2*Range[2, m]-1}, Reverse[Map[Total[r-#] &, Select[Subsets[Range[2, 2*m-1], {m-1}], Min[r-#] >= 0 &]]]];
A236855list[6] (* Generates C(6) terms *) (* Paolo Xausa, Feb 19 2024 *)

(define (A236855 n) (apply + (A239903raw n)))
(define (A239903raw n) (if (zero? n) (list) (let loop ((n n) (row (- (A081288 n) 1)) (col (- (A081288 n) 2)) (srow (- (A081288 n) 2)) (catstring (list 0))) (cond ((or (zero? row) (negative? col)) (reverse! (cdr catstring))) ((> (A009766tr row col) n) (loop n srow (- col 1) (- srow 1) (cons 0 catstring))) (else (loop (- n (A009766tr row col)) (+ row 1) col srow (cons (+ 1 (car catstring)) (cdr catstring))))))))
;; Alternative definition:
(define (A236855 n) (let ((x (A071155 (A081291 n)))) (- (A034968 x) (A060130 x))))

a(n) = A034968(x) - A060130(x), where x = A071155(A081291(n)).
For up to n = A000108(11)-2 = 58784, a(n) = A007953(A239903(n)).
Catalan numbers, A000108, give the positions of ones, and the n-th triangular number occurs for the first time at the position immediately before that, i.e., a(A001453(n)) = A000217(n-1).
For each n, a(n) >= A000217(A236859(n)).

A236859 The length of the initial ascent 123... in the n-th Catalan numeral, A239903(n).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2

Offset: 0

Antti Karttunen, Apr 18 2014

nonn

			A239903(1) = 1, thus a(1) = 1.
A239903(2) = 10, thus a(2) = 1.
A239903(4) = 12, thus a(4) = 2.
A239903(39) = 1232, thus a(39) = 3.
A239903(58784) = 1234567899, thus a(58784) = 9.
Note that although the range of validity of A239903 is inherently limited by the decimal representation employed, it doesn't matter here: We have a(58785) = 10, as the corresponding 58785th Catalan String is [1,2,3,4,5,6,7,8,9,10], even though A239903 cannot represent that unambiguously.

Antti Karttunen, Table of n, a(n) for n = 0..16796

Cf. A239903, A236855, A081291, A126307.

(define (A236859 n) (if (zero? n) n (- (A126307 (A081291 n)) 1)))

a(0) = 0, and for n>=1, a(n) = A126307(A081291(n))-1.

Each n occurs for the first time (as a record) at the position (C_{n+1})-1, so we have a(A001453(n+1)) = n for all n.

A323224 A(n, k) = [x^k] C^nx/(1 - x) where C = 2/(1 + sqrt(1 - 4x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 4, 1, 0, 1, 4, 8, 9, 1, 0, 1, 5, 13, 22, 23, 1, 0, 1, 6, 19, 41, 64, 65, 1, 0, 1, 7, 26, 67, 131, 196, 197, 1, 0, 1, 8, 34, 101, 232, 428, 625, 626, 1, 0, 1, 9, 43, 144, 376, 804, 1429, 2055, 2056, 1

Offset: 0

Peter Luschny, Jan 24 2019

Equals A096465 when the leading column (k = 0) is removed. - Georg Fischer, Jul 26 2023

			The square array starts:
   [n\k]  0  1   2   3    4     5     6      7       8       9
    ---------------------------------------------------------------
    [0]   0, 1,  1,  1,   1,    1,    1,     1,      1,      1, ... A057427
    [1]   0, 1,  2,  4,   9,   23,   65,   197,    626,   2056, ... A014137
    [2]   0, 1,  3,  8,  22,   64,  196,   625,   2055,   6917, ... A014138
    [3]   0, 1,  4, 13,  41,  131,  428,  1429,   4861,  16795, ... A001453
    [4]   0, 1,  5, 19,  67,  232,  804,  2806,   9878,  35072, ... A114277
    [5]   0, 1,  6, 26, 101,  376, 1377,  5017,  18277,  66727, ... A143955
    [6]   0, 1,  7, 34, 144,  573, 2211,  8399,  31655, 118865, ...
    [7]   0, 1,  8, 43, 197,  834, 3382, 13378,  52138, 201364, ...
    [8]   0, 1,  9, 53, 261, 1171, 4979, 20483,  82499, 327656, ...
    [9]   0, 1, 10, 64, 337, 1597, 7105, 30361, 126292, 515659, ...
.
Triangle given by ascending antidiagonals:
    0;
    0, 1;
    0, 1, 1;
    0, 1, 2,  1;
    0, 1, 3,  4,   1;
    0, 1, 4,  8,   9,   1;
    0, 1, 5, 13,  22,  23,   1;
    0, 1, 6, 19,  41,  64,  65,   1;
    0, 1, 7, 26,  67, 131, 196, 197,   1;
    0, 1, 8, 34, 101, 232, 428, 625, 626, 1;
.
The difference table of a column successively gives the preceding columns, here starting with column 6.
col(6) = 1, 65, 196, 428, 804, 1377, 2211, 3382, 4979, 7105, ...
col(5) =    64, 131, 232, 376,  573,  834, 1171, 1597, 2126, ...
col(4) =         67, 101, 144,  197,  261,  337,  426,  529, ...
col(3) =              34,  43,   53,   64,   76,   89,  103, ...
col(2) =                    9,   10,   11,   12,   13,   14, ...
col(1) =                          1,    1,    1,    1,    1, ...
col(0) =                                0,    0,    0,    0, ...
.
Example for the sum formula: C(0) = 1, C(1) = 1, C(2) = 2 and C(3) = 5.
X(3, 4) = {{0,0,0}, {0,0,1}, {0,1,0}, {1,0,0}, {0,0,2}, {0,1,1}, {0,2,0}, {1,0,1},
{1,1,0}, {2,0,0}, {0,0,3}, {0,1,2}, {0,2,1}, {0,3,0}, {1,0,2}, {1,1,1}, {1,2,0},
{2,0,1}, {2,1,0}, {3,0,0}}. T(3,4) = 1+1+1+1+2+1+2+1+1+2+5+2+2+5+2+1+2+2+2+5 = 41.

The coefficients of the polynomials generating the columns are in A323233.
Sums of antidiagonals and row 1 are A014137. Main diagonal is A242798.
Rows: A057427 (n=0), A014137 (n=1), A014138 (n=2), A001453 (n=3), A114277 (n=4), A143955 (n=5).
Columns: A000027 (k=2), A034856 (k=3), A323221 (k=4), A323220 (k=5).
Similar array based on central binomials is A323222.
Cf. A096465.

Row := proc(n, len) local C, ogf, ser; C := (1-sqrt(1-4*x))/(2*x);
ogf := C^n*x/(1-x); ser := series(ogf, x, (n+1)*len+1);
seq(coeff(ser, x, j), j=0..len) end:
for n from 0 to 9 do Row(n, 9) od;
# Alternatively by recurrence:
B := proc(n, k) option remember; if n <= 0 or k < 0 then 0
elif n = k then 1 else B(n-1, k) + B(n, k-1) fi end:
A := (n, k) -> B(n + k, k): seq(lprint(seq(A(n, k), k=0..9)), n=0..9);

(* Illustrating the sum formula, not efficient. *) T[0, K_] := Boole[K != 0];
T[N_, K_] := Module[{}, r[n_, k_] := FrobeniusSolve[ConstantArray[1, n], k];
X[n_] := Flatten[Table[r[N, j], {j, 0, n - 1}], 1];
Sum[Product[CatalanNumber[m[[i]]], {i, 1, N}], {m , X[K]}]];
Trow[n_] := Table[T[n, k], {k, 0, 9}]; Table[Trow[n], {n, 0, 9}]

For n>0 and k>0 let X(n, k) denote the set of all tuples of length n with elements from {0, ..., k-1} with sum < k. Let C(m) denote the m-th Catalan number. Then: A(n, k) = Sum_{(j1,...,jn) in X(n, k)} C(j1)*C(j2)*...*C(jn).

A(n, k) = T(n + k, k) with T(n, k) = T(n-1, k) + T(n, k-1) with T(n, k) = 0 if n <= 0 or k < 0 and T(n, n) = 1.

A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 13, 1, 1, 41, 9, 1, 1, 131, 59, 11, 1, 1, 428, 344, 88, 15, 1, 1, 1429, 1906, 634, 146, 23, 1, 1, 4861, 10345, 4389, 1231, 280, 39, 1, 1, 16795, 55901, 30006, 9835, 2763, 602, 71, 1, 1, 58785, 303661, 205420, 77178, 25014, 6967, 1408, 135, 1

Offset: 0

Alois P. Heinz, May 28 2017

			T(4,0) = 1: 1234.
T(4,1) = 13: 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
T(4,2) = 1: 13|2|4.
T(5,2) = 9: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5.
T(6,3) = 11: 1245|3|6, 1346|2|5, 134|26|5, 134|2|56, 134|2|5|6, 145|23|6, 145|2|36, 145|2|3|6, 14|25|3|6, 15|24|3|6, 1|245|3|6.
T(6,4) = 1: 1345|2|6.
T(7,4) = 15: 12456|3|7, 13457|2|6, 1345|27|6, 1345|2|67, 1345|2|6|7, 1456|23|7, 1456|2|37, 1456|2|3|7, 145|26|3|7, 146|25|3|7, 14|256|3|7, 156|24|3|7, 15|246|3|7, 16|245|3|7, 1|2456|3|7.
Triangle T(n,k) begins:
  1;
  1;
  1,    1;
  1,    4;
  1,   13,     1;
  1,   41,     9,    1;
  1,  131,    59,   11,    1;
  1,  428,   344,   88,   15,   1;
  1, 1429,  1906,  634,  146,  23,  1;
  1, 4861, 10345, 4389, 1231, 280, 39, 1;
  ...

Alois P. Heinz, Rows n = 0..20, flattened
Wikipedia, Partition of a set

Columns k=0-1 give: A000012, A001453.
Row sums give A000110.
Cf. A168415, A287213, A287215, A287416, A287641.

b:= proc(n, l) option remember; `if`(n=0 or l=[], 1, add(b(n-1,
      [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
    end:
T:= (n, k)-> `if`(k=0, 1, b(n, [0$k])-b(n, [0$k-1])):
seq(seq(T(n, k), k=0..max(n/2, n-2)), n=0..12);

b[n_, l_] := b[n, l] = If[n == 0 || l == {}, 1, Sum[b[n-1, Append[Table[ Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
T[n_, k_] := If[k == 0, 1, b[n, Table[0, k]] - b[n, Table[0, k - 1]]];
Table[T[n, k], {n, 0, 12}, { k, 0, Max[n/2, n - 2]}] // Flatten (* Jean-François Alcover, May 22 2018, translated from Maple *)

T(n,k) = A287641(n,k) - A287641(n,k-1) for k>0, T(n,0) = 1.

T(n+4,n+1) = A168415(n) for n>0.

A085180 Array A(x,y) giving the position of the y-th x in A007001 listed by rising antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 4, 6, 14, 10, 7, 8, 42, 28, 13, 9, 11, 132, 84, 37, 19, 12, 15, 429, 264, 112, 41, 24, 16, 17, 1430, 858, 354, 126, 56, 27, 18, 20, 4862, 2860, 1155, 402, 131, 70, 33, 21, 22, 16796, 9724, 3861, 1320, 422, 174, 79, 36, 23, 25, 58786, 33592, 13156, 4433

Offset: 1

Antti Karttunen, Jun 18 2003

Read by upwards antidiagonals as A(1,1), A(2,1), A(1,2), A(3,1), A(2,2), A(1,3), etc.

			In A007001 each n occurs for the first time at position Cat(n), thus A(x,1) (the first row) is A000108.

Index entries for sequences that are permutations of the natural numbers

Variant: A085178.
Inverse permutation: A085181.
First row: A000108, second row: A068875 apart from initial terms, first column: A085197, central diagonal: A001453.

A096465 Triangle (read by rows) formed by setting all entries in the first column and in the main diagonal ((i,i) entries) to 1 and the rest of the entries by the recursion T(n, k) = T(n-1, k) + T(n, k-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 8, 9, 1, 1, 5, 13, 22, 23, 1, 1, 6, 19, 41, 64, 65, 1, 1, 7, 26, 67, 131, 196, 197, 1, 1, 8, 34, 101, 232, 428, 625, 626, 1, 1, 9, 43, 144, 376, 804, 1429, 2055, 2056, 1, 1, 10, 53, 197, 573, 1377, 2806, 4861, 6917, 6918, 1, 1, 11, 64, 261, 834, 2211, 5017, 9878, 16795, 23713, 23714, 1

Offset: 0

Gerald McGarvey, Aug 12 2004

The third column is A034856 (binomial(n+1, 2) + n-1).
The row sums are A014137 (partial sums of Catalan numbers (A000108)).
The "1st subdiagonal" ((i+1,i) entries) are also A014137.
The "2nd subdiagonal" ((i+2,i) entries) is A014138 ( Partial sums of Catalan numbers (starting 1,2,5,...)).
The "3rd subdiagonal" ((i+3,i) entries) is A001453 (Catalan numbers - 1.)
This is the reverse of A091491 - see A091491 for more information. The sequence of antidiagonal sums gives A124642. - Gerald McGarvey, Dec 09 2006

			Triangle begins as:
  1;
  1, 1;
  1, 2,  1;
  1, 3,  4,  1;
  1, 4,  8,  9,   1;
  1, 5, 13, 22,  23,   1;
  1, 6, 19, 41,  64,  65,   1;
  1, 7, 26, 67, 131, 196, 197, 1;

Reinhard Zumkeller, Rows n=0..150 of triangle, flattened

Cf. A000108, A001453, A014137, A014138, A034856.

Cf. A006134, A024718, A030237, A078478, A091491, A100066, A105848, A124642.

a096465 n k = a096465_tabl !! n !! k
a096465_row n = a096465_tabl !! n
a096465_tabl = map reverse a091491_tabl
-- Reinhard Zumkeller, Jul 12 2012

A096465:= func< n,k | k eq n select 1 else (n-k)*(&+[Binomial(n+k-2*j, n-j)/(n+k-2*j): j in [0..k]]) >;
[A096465(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 30 2021

A096465:= (n,k)-> `if`(k=n, 1, (n-k)*add(binomial(n+k-2*j, n-j)/(n+k-2*j), j=0..k));
seq(seq(A096465(n,k), k=0..n), n=0..12) # G. C. Greubel, Apr 30 2021

T[, 0]= 1; T[n, n_]= 1; T[n_, m_]:= T[n, m]= T[n-1, m] + T[n, m-1]; T[n_, m_] /; n < 0 || m > n = 0; Table[T[n, m], {n, 0, 12}, {m, 0, n}]//Flatten (* Jean-François Alcover, Dec 17 2012 *)

def A096465(n,k): return 1 if (k==n) else (n-k)*sum( binomial(n+k-2*j, n-j)/(n+k-2*j) for j in (0..k))
flatten([[A096465(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 30 2021

From G. C. Greubel, Apr 30 2021: (Start)
T(n, k) = (n-k) * Sum_{j=0..k} binomial(n+k-2*j, n-j)/(n+k-2*j) with T(n,n) = 1.
T(n, k) = A091491(n, n-k).
Sum_{k=0..n} T(n,k) = Sum_{j=0..n} A000108(j) = A014137(n). (End)

Offset changed by Reinhard Zumkeller, Jul 12 2012

cp's OEIS Frontend

A289616 A246604 (Catalan(n)-n) with initial terms 1,0,0,2,10 changed to 1,1,1,2,11.

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A289652 Catalan numbers - 2 (A120304) with first three terms changed to 1,1,1.

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A289653 Catalan numbers - 2 (A120304) with first four terms changed to 1,1,1,4.

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A289654 Related to number of mesh patterns of length 2 that avoid the pattern 321.

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A236855 a(n) is the sum of digits in A239903(n).

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A236859 The length of the initial ascent 123... in the n-th Catalan numeral, A239903(n).

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A323224 A(n, k) = [x^k] C^n*x/(1 - x) where C = 2/(1 + sqrt(1 - 4*x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.

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Maple

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A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

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A085180 Array A(x,y) giving the position of the y-th x in A007001 listed by rising antidiagonals.

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A323224 A(n, k) = [x^k] C^nx/(1 - x) where C = 2/(1 + sqrt(1 - 4x)), square array read by ascending antidiagonals with n >= 0 and k >= 0.