A000245 -id:A000245 - cp's OEIS frontend

A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

1, 11, 165, 2860, 53900, 1072797, 22188859, 472214600, 10273141395, 227440759700, 5107663394691, 116068178638776, 2664012608972000, 61668340817988135, 1438101958237201950, 33753007927148177360, 796704536753910327114

Offset: 0

Tim Fulford, Oct 04 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 10, r = 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A234573.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10).

[11*Binomial(10*n+11,n)/(10*n+11) : n in [0..20]]; // Vincenzo Librandi, Jan 10 2014

Table[11/(10 n + 11) Binomial[10 n + 11, n], {n, 0, 40}] (* Vincenzo Librandi, Jan 10 2014 *)

a(n) = 11*binomial(10*n+11,n)/(10*n+11);

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/11))^11+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 10, r = 11.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^11), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/11) is the o.g.f. for A059968. (End)
D-finite with recurrence: 81*n*(9*n+11)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+10)*(3*n+1)*(9*n+5)*(9*n+7)*a(n) -800*(10*n+1)*(5*n+1)*(10*n+3)*(5*n+2)*(2*n+1)*(5*n+3)*(10*n+7)*(5*n+4)*(10*n+9)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

Corrected by Vincenzo Librandi, Jan 10 2014

A232265 a(n) = 10binomial(9n + 10, n)/(9*n + 10).

Original entry on oeis.org

1, 10, 135, 2100, 35475, 632502, 11714745, 223198440, 4346520750, 86128357150, 1731030945644, 35202562937100, 723029038312230, 14976976398326250, 312522428615310000, 6563314391270476752, 138617681440915119975, 2942332729799060033100, 62735156704285184848950

Offset: 0

Tim Fulford, Dec 28 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 9, r = 10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A143554, A234505, A234506, A234507, A234508, A234509, A234510, A234513.

Cf. A062994, A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A229963 (k = 11).

[10*Binomial(9*n+10, n)/(9*n+10): n in [0..30]];

Table[10 Binomial[9 n + 10, n]/(9 n + 10), {n, 0, 30}]

PARI
```
a(n) = 10*binomial(9*n+10,n)/(9*n+10);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/10))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 9, r = 10.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^10), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/10) is the o.g.f. for A062994. (End)
D-finite with recurrence: 128*n*(8*n+3)*(4*n+3)*(8*n+9)*(2*n+1)*(8*n+7)*(4*n+5)*(8*n+5)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

A239903 List of Restricted-Growth Strings a_{k-1}a_{k-2}...a_{2}a_{1}, with k=2 and a_1 in {0,1} or k>2, a_{k-1}=1 and a_{j+1}>=1+a_j, for k-1>j>0.

Original entry on oeis.org

0, 1, 10, 11, 12, 100, 101, 110, 111, 112, 120, 121, 122, 123, 1000, 1001, 1010, 1011, 1012, 1100, 1101, 1110, 1111, 1112, 1120, 1121, 1122, 1123, 1200, 1201, 1210, 1211, 1212, 1220, 1221, 1222, 1223, 1230, 1231, 1232, 1233, 1234, 10000, 10001, 10010, 10011

Offset: 0

N. J. A. Sloane, Apr 06 2014

We write the nonnegative integers as restricted growth strings (so called by J. Arndt in his book fxtbook.pdf, p. 325) in such a way that the Catalan numbers (cf. A000108) are expressed: 1=1, 10=2, 100=5, 1000=14, etc., 10...0 (with k zeros) = the k-th Catalan number. Once the entries of a restricted-growth string grow above 9, one would need commas or parentheses, say, to separate those entries. See Dejter (2017) for the precise definition.
In the paper "A system of numeration for middle-levels", restricted growth strings (RGSs) are defined as sequences that begin with either 0 or 1, with each successive number to the right being at least zero and at most one greater than its immediate left neighbor. Moreover, apart from case a(0), the RGSs are finite integer sequences of restricted growth which always start with 1 as their first element b_1 in position 1, and from then on, each successive element b_{i+1} in the sequence is restricted to be in range [0,(b_i)+1].
This sequence gives all such finite sequences in size-wise and lexicographic order, represented as decimal numbers by concatenating the integers of such finite sequences (e.g., from [1,2,0,1] we get 1201). The 58784th such sequence is [1, 2, 3, 4, 5, 6, 7, 8, 9, 9], thus a(58784) = 1234567899, after which comes the first RGS, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], where an element larger than 9 is present, which means that the decimal system employed here is unambiguous only up to n=58784. Note that 58785 = A000108(11)-1.
Also, if one considers Stanley's interpretation (u) of Catalan numbers, "sequences of a_1, a_2, ..., a_n of integers such that a_1 = 0 and 0 <= a_{i+1} <= a_{i} + 1" (e.g., 000, 001, 010, 011, 012 for C_3), and discards their initial zero, then one has a bijective correspondence with Dejter's RGSs of one element shorter length, which in turn are in bijective correspondence with the first C_n terms of this sequence (by discarding any leading zeros), from a(0) to a(C_n - 1). From this follows that the k-th Catalan number, A000108(k) (k>0), is represented in this system as 1 followed by k-1 zeros: a(1)=1, a(2)=10, a(5)=100, a(14)=1000, etc., and also that there exist exactly A000245(k) RGSs of length k.
Note how this differs from other number representations utilizing Catalan numbers, A014418 and A244159, in that while the latter are base-systems, where a simple weighted Sum_{k} digit(k)*C(k) recovers the natural number n (which the n-th numeral of such system represents), in contrast here it is the sum of appropriate terms in Catalan's Triangle (A009766, A030237), obtained by unranking a unique instance of a certain combinatorial structure (one of the Catalan interpretations), that gives a correspondence with a unique natural number. (Cf. also A014486.)
This sequence differs from "Semigreedy Catalan Representation", A244159, for the first time at n=10, where a(10) = 120, while A244159(10) = 121. That is also the first position where A244158(a(n)) <> n.
Please see Dejter's preprint for a more formal mathematical definition and how this number system is applied in relation to Havel's Conjecture on the existence of Hamiltonian cycles in the middle-levels graphs.
a(n) is given by the concatenation (with leading zeros removed) of the terms of row n + 23714 of A370222. - Paolo Xausa, Feb 17 2024

			Catalan's Triangle T(row,col) = A009766 begins with row n=0 and 0<=col<=n as:
  Row 0: 1
  Row 1: 1, 1
  Row 2: 1, 2,  2
  Row 3: 1, 3,  5,  5
  Row 4: 1, 4,  9, 14, 14
  Row 5: 1, 5, 14, 28, 42,  42
  Row 6: 1, 6, 20, 48, 90, 132, 132
  (the leftmost diagonal of 1s is "column 0").
  ...
For example, for n=38, we find that A081290(38)=14, which occurs on row A081288(n)-1 = 4, in columns A081288(n)-1 and A081288(n)-2, i.e., as T(4,4) and T(4,3). Thus we subtract 38-14 to get 24, and we see that the next term downward on the same diagonal, 28, is too large to accommodate into the same sum, so we go one diagonal up, starting now from T(3,2) = 5. This fits in, so we now have 24 - 5 = 19, and also the next term on the same diagonal, T(4,2) = 9, fits in, so we now have 19-9 = 10. The next term on the same diagonal, T(5,2) = 14, would not fit in anymore, so we rewind ourselves back to penultimate column, but one step up from where we started on this diagonal, so T(2,1) = 2, which fits in, 10 - 2 = 8, also the next one T(3,1) = 3, 8 - 3 = 5, and the next one T(4,1) = 4, 5 - 4 = 1, after which comes T(5,1) = 5 > 1, thus we jump to T(1,0) = 1, 1-1 = 0, and T(2,0)=1 would not fit anymore, thus next time the row would be zero, and the algorithm is ready with 1 (14), 2 (5+9), 3 (2+3+4) and 1 (1) terms collected, whose total sum 14+5+9+2+3+4+1 = 38, thus a(38) = 1231.
For n=20, the same algorithm results in 1 (14), 1 (5), 0 (not even the first tentative term T(2,1) = 2 from the column 1 would fit, so it is skipped), and from one row higher we get the needed 1 (1), so the total sum of these is 14+5+0+1 = 20, thus a(20) = 1101.

D. E. Knuth, The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, third edition, Addison-Wesley, 1977, p. 192.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999, Exercise 19, interpretation (u).

Antti Karttunen, Table of n, a(n) for n = 0..16796
Joerg Arndt, Matters Computational: Ideas, Algorithm, Source Code, Springer, 2011; freely downloadable here: Fxtbook.
Georg Cantor, Über die einfachen Zahlensysteme, Zeitschrift fur Mathematik und Physik, 14 (1869), 121-128.
Italo J. Dejter, A system of numeration for middle levels, arXiv:1012.0995 [math.CO], (2014).
Italo J. Dejter, The role of restricted growth strings in the two middle levels of the Boolean lattice B_(2k+1), University of Puerto Rico, 2018.
Italo J. Dejter, Reinterpreting Mütze's theorem via natural enumeration of ordered rooted trees, arXiv:1911.02100 [math.CO], (2019).
Italo J. Dejter, A numeral system for the middle-levels graphs, Elec. J. Graph Theory and Applications (2021) Vol. 9, No. 1, 137-156. See p. 138.
Italo J. Dejter, Edge-supplementary arc factorizations of odd graphs, uniform 2-factors and Hamilton cycles, arXiv:2203.05326 [math.CO], 2022.
Italo J. Dejter, Integer sequence of Dyck-path single changes, arXiv:2209.11122 [math.CO], 2022.
Italo J. Dejter, A Dyck-word tree that controls all odd graphs, arXiv:2306.14249 [math.CO], 2023.
Aviezri S. Fraenkel, The Use and Usefulness of Numeration Systems, Inf. Comput., 81(1), (1989), 46-61.
Aviezri S. Fraenkel, Systems of numeration, IEEE Symposium on Computer Arithmetic, (1983), 37-42.
Aviezri S. Fraenkel, Systems of numeration, The American Mathematical Monthly, Vol. 92, No. 2 (Feb., 1985), pp. 105-114.
Antti Karttunen, Catalan ranking and unranking functions, OEIS Wiki.
R. P. Stanley, Exercises on Catalan and Related Numbers, interpretation (u) of Catalan numbers, on page 224 (page 4/27 in the November 1 2000 version of the PDF-file).

Cf. A000108 (Catalan numbers), A000245 (their first differences), A009766 (Catalan's triangle), A236855 (the sum of elements in k-th RGS), A236859 (for n>=1, gives the length of the initial ascent 123... in term a(n)), A244159 (different kinds of Catalan number systems).
Other Catalan combinatorial structures represented as integer sequences: A014486/A063171: Dyck words, parenthesizations, etc., A071156/A071158: Similar restricted words encoded with help of A007623 (Integers written in factorial base), A071153/A079436 (Łukasiewicz words).
Cf. A007001, A007623, A014418, A071154, A071159, A081288, A081290, A081291, A082853, A126307, A244155, A244156, A244157, A244158, A370222.

function CatalanNumerals(z)
    z == 0 && return 0
    f(n) = factorial(n)
    t(j, k) = div(f(k+j)*(k-j+1), f(j)*f(k+1))
    k, i = 2, 0
    while z >= t(i, i + 1) i += 1 end
    dig = fill(0, i); dig[1] = 1
    x = z - t(i - 1, i)
    m = i - 1
    while x > 0
        w, s, p = 0, 0, 0
        while w <= x
            p = w
            w += t(m - 1, m + s)
            s += 1
        end
        dig[k] = s - 1
        m -= 1; k += 1; x -= p
    end
    s = ""; for d in dig s *= string(d) end
    parse(Int, s)
end
[CatalanNumerals(n) for n in 0:42] |> println # Peter Luschny, Nov 10 2019

function [ c ] = catrep(z)
i=0; x=0; y=0; s=0;
while z>=(factorial(2*i+1)*(2))/(factorial(i)*factorial(i+2))
i=i+1;
end
y=(factorial(2*i-1)*(2))/(factorial(i-1)*factorial(i+1));
a=zeros(1,i); a(1,1)=1; k=2; x=z-y; m=1;
while x>0
w=0; s=0; p=0;
while w<=x
p=w;
w=w+(factorial(2*i-2*m+s-1)*(s+2))/(factorial(i-1-m)*factorial(i-m+s+1));
s=s+1;
end
m=m+1; a(1,k)=s-1; k=k+1; x=x-p;
end
a
end

A239903full = With[{r = 2*Range[2, 11]-1}, Reverse[Map[FromDigits[r-#] &, Rest[Select[Subsets[Range[2, 21], {10}, 125477], Min[r-#] >= 0 &]]]]];
A239903full[[;;100]] (* Paolo Xausa, Feb 17 2024 *)

define (t(j,k), (factorial(k+j)*(k-j+1))/(factorial(j)*factorial(k+1)));
i:0;
x:19;
z:0;y:0;s:0;
while x>=t(i,i+1) do (i:i+1);
y:t(i-1,i);a:zeromatrix(1,i);a[1,1]:1;k:2;z:x-y;m:1;
while (z>0) do (
w:0,s:0,p=0,
while (w<=z) do (
p:w,
w:w+t(i-1-m,i-m+s),
s:s+1
),
m:m+1,
a[1,k]:s-1,k:k+1,
z:z-p
);
print(a);

\\ Valid for n<58786 (=A000108(11)).
nxt(w)=if(w[1]==#w, vector(#w+1, i, i>#w), my(k=1); while(w[k]>w[k+1], w[k]=0; k++); w[k]++; w)
seq(n)={my(a=vector(n), w=[1]); a[1]=0; for(i=2, #v, a[i]=fromdigits(Vecrev(w)); w=nxt(w)); a} \\ Andrew Howroyd, Jan 24 2023

(define (A239903_only_upto_16794 n) (if (zero? n) n (A235049 (A071159 (A081291 n))))) ;; Gives correct results only up to 16794.
;; The following gives correct results all the way up to n=58784.
(define (A239903 n) (baselist-as-decimal (A239903raw n)))
(definec (A239903raw n) (if (zero? n) (list) (let loop ((n n) (row (A244160 n)) (col (- (A244160 n) 1)) (srow (- (A244160 n) 1)) (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))))))))
(define (baselist-as-decimal lista) (baselist->n 10 lista))
(define (baselist->n base bex) (let loop ((bex bex) (n 0)) (cond ((null? bex) n) (else (loop (cdr bex) (+ (* n base) (car bex)))))))
;; From Antti Karttunen, Apr 14-19 2014

To find an RGS corresponding to natural number n, one first finds a maximum row index k such that T(k,k-1) <= n in the Catalan Triangle (A009766) illustrated in the Example section. Note that as the last two columns of this triangle consist of Catalan numbers (that is, T(k,k-1) = T(k,k) = A000108(k)), it means that the first number to be subtracted from n is A081290(n) which occurs as a penultimate element of the row A081288(n)-1, in the column A081288(n)-2. The unranking algorithm then proceeds diagonally downwards, keeping the column index the same, and incrementing the row index, as long as it will encounter terms such that their total sum stays less than or equal to n.
If the total sum of encountered terms on that diagonal would exceed n, the algorithm jumps back to the penultimate column of the triangle, but one row higher from where it started the last time, and again starts summing the terms as long as the total sum stays <= n.
When the algorithm eventually reaches either row zero or column less than zero, the result will be a list of numbers, each element being the number of terms summed from each diagonal, so that the diagonal first traversed appears as the first 1 (as that first diagonal will never allow more than one term), and the number of terms summed from the last traversed diagonal appears the last number in the list. These lists of numbers are then concatenated together as decimal numbers.
These steps can also be played backwards in order to recover the corresponding decimal integer n from such a list of numbers, giving a "ranking function" which will be the inverse to this "unranking function".
For n=1..16794 (where 16794 = A000108(10)-2), a(n) = A235049(A071159(A081291(n))). - Antti Karttunen, Apr 14 2014
Alternative, simpler description of the algorithm from Antti Karttunen, Apr 21 2014: (Start)
Consider the following square array, which is Catalan triangle A009766 without its rightmost, "duplicate" column, appropriately transposed (cf. also tables A030237, A033184 and A054445):
   Row| Terms on that row
   ---+--------------------------
    1 |   1   1   1    1    1 ...
    2 |   2   3   4    5    6 ...
    3 |   5   9  14   20   27 ...
    4 |  14  28  48   75  110 ...
    5 |  42  90 165  275  429 ...
    6 | 132 297 572 1001 1638 ...
To compute the n-th RGS, search first for the greatest Catalan number C_k which is <= n (this is A081290(n), found as the first term of row A081288(n)-1). Then, by a greedy algorithm, select from each successive row (moving towards the top of table) as many terms from the beginning of that row as will still fit into n, subtracting them from n as you go. The number of terms selected from the beginning of each row gives each element of the n-th RGS, so that the number of terms selected from the topmost row (all 1's) appears as its last element.
(End)

Description, formula and examples edited/rewritten by Italo J Dejter, Apr 13 2014 and Antti Karttunen, Apr 18 2014

A026012 Second differences of Catalan numbers A000108.

Original entry on oeis.org

1, 2, 6, 19, 62, 207, 704, 2431, 8502, 30056, 107236, 385662, 1396652, 5088865, 18642420, 68624295, 253706790, 941630580, 3507232740, 13105289370, 49114150020, 184560753390, 695267483664, 2625197720454, 9933364416572, 37660791173152, 143048202990504

Offset: 0

N. J. A. Sloane, Clark Kimberling

nonn

Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = s(2n) = 2.
Number of Dyck paths of semilength n+2 with no initial and no final UD's. Example: a(2)=6 because the only Dyck paths of semilength 4 with no initial and no final UD's are UUDUDUDD, UUDUUDDD, UUUDDUDD, UUUDUDDD, UUDDUUDD, UUUUDDDD. - Emeric Deutsch, Oct 26 2003
Number of branches of length 1 starting from the root in all ordered trees with n+1 edges. Example: a(1)=2 because the tree /\ has two branches of length 1 starting from the root and the path-tree of length 2 has none. a(n) = Sum_{k=0..n+1} (k*A127158(n+1,k)). - Emeric Deutsch, Mar 01 2007
Number of staircase walks from (0,0) to (n,n) that never cross y=x+2. Example: a(3) = 19 because up,up,up,right,right,right is not allowed but the other binomial(6,3)-1 = 19 paths are. - Mark Spindler, Nov 11 2012
Number of standard Young tableaux of skew shape (n+2,n)/(2), for n>=2. - Ran Pan, Apr 07 2015

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see pp. 188, 196).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
Reinis Cirpons, James East, and James D. Mitchell, Transformation representations of diagram monoids, arXiv:2411.14693 [math.RA], 2024. See pp. 3, 33.
Ryota Inagaki, Tanya Khovanova, and Austin Luo, On Chip-Firing on Undirected Binary Trees, Ann. Comb. (2025). See p. 15.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016. See Appendix B2.

T(2n, n), where T is the array defined in A026009.

Cf. A000108, A000245, A000344, A039599, A059346, A127158, A342912.

Differences[Table[CatalanNumber[n], {n, 0, 28}], 2] (* Jean-François Alcover, Sep 28 2012 *)
Table[Binomial[2n,n]-Binomial[2n,n-3],{n,0,26}] (* Mark Spindler, Nov 11 2012 *)

a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)); /* Joerg Arndt, Aug 19 2012 */

Expansion of (1+x^1*C^3)*C^1, where C = (1-(1-4*x)^(1/2))/(2*x) is g.f. for Catalan numbers, A000108.
a(n) = 3*(3*n^2+3*n+2)*binomial(2*n, n)/((n+1)*(n+2)*(n+3)). - Emeric Deutsch, Oct 26 2003
a(n) = Sum_{k=0..2} A039599(n,k) = A000108(n) + A000245(n) + A000344(n). - Philippe Deléham, Nov 12 2008
a(n) = binomial(2*n,n)/(n+1)*hypergeom([-2,n+1/2],[n+2],4). - Peter Luschny, Aug 15 2012
a(n) = binomial(2*n,n) - binomial(2n,n-3). - Mark Spindler, Nov 11 2012
D-finite with recurrence (n+3)*a(n) + (-5*n-6)*a(n-1) + 2*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Jun 20 2013
E.g.f.: exp(2*x)*(BesselI(0,2*x) - BesselI(3,2*x)). - Ilya Gutkovskiy, Feb 28 2017
Sum_{n>=0} a(n)/4^n = 6. - Amiram Eldar, Jul 10 2023
a(n) = C(n+2)+C(n)-2*C(n+1), C = A000108. - Alois P. Heinz, Apr 02 2025
Binomial transform of A342912. - Mélika Tebni, Apr 05 2025

A067323 Catalan triangle A028364 with row reversion.

Original entry on oeis.org

1, 2, 1, 5, 3, 2, 14, 9, 7, 5, 42, 28, 23, 19, 14, 132, 90, 76, 66, 56, 42, 429, 297, 255, 227, 202, 174, 132, 1430, 1001, 869, 785, 715, 645, 561, 429, 4862, 3432, 3003, 2739, 2529, 2333, 2123, 1859, 1430, 16796, 11934, 10504, 9646, 8986, 8398, 7810, 7150, 6292, 4862

Offset: 0

Wolfdieter Lang, Feb 05 2002

a(N,p) equals X_{N}(N+1,p) := T_{N,p} for alpha= 1 =beta and N>=p>=1 in the Derrida et al. 1992 reference. The one-point correlation functions A000108(n)%20(Catalan)%20in%20this%20reference.%20See%20also%20the%20Derrida%20et%20al.%201993%20reference.%20In%20the%20Liggett%201999%20reference%20mu">{N} for alpha= 1 =beta equal a(N,K)/C(N+1) with C(n)=A000108(n) (Catalan) in this reference. See also the Derrida et al. 1993 reference. In the Liggett 1999 reference mu{N}{eta:eta(k)=1} of prop. 3.38, p. 275 is identical with _{N} and rho=0 and lambda=1.
Identity for each row n>=1: a(n,m)+a(n,n-m+1)= C(n+1), with C(n+1)=A000108(n+1)(Catalan) for every m=1..floor((n+1)/2). E.g., a(2k+1,k+1)=C(2*(k+1)).
The first column sequences (diagonals of A028364) are: A000108(n+1), A000245, A067324-6 for m=0..4.

			Triangle begins:
     1;
     2,    1;
     5,    3,    2;
    14,    9,    7,    5;
    42,   28,   23,   19,   14;
   132,   90,   76,   66,   56,   42;
   429,  297,  255,  227,  202,  174,  132;
  1430, 1001,  869,  785,  715,  645,  561,  429;
  4862, 3432, 3003, 2739, 2529, 2333, 2123, 1859, 1430;
  ...

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (19) - (23), p. 672.
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eqs. (43), (44), pp. 1501-2 and eq.(81) with eqs.(80) and (81).
T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, 1999, pp. 269, 275.
G. Schuetz and E. Domany, Phase Transitions in an Exactly Soluble one-Dimensional Exclusion Process, J. Stat. Phys. 72 (1993) 277-295, eq. (2.18), p. 283, with eqs. (2.13)-(2.15).

Alois P. Heinz, Rows n = 0..140, flattened
Steve Butler, Kimberly Hadaway, Victoria Lenius, Preston Martens, and Marshall Moats, Lucky cars and lucky spots in parking functions, arXiv:2412.07873 [math.CO], 2024. See p. 14.
Wolfdieter Lang, First 10 rows.

Cf. A001700 (row sums).
Cf. A039598, A039599.
T(2n,n) gives A201205.

b:= proc(n, i) option remember; `if`(n=0, 1, add(
      expand(b(n-1, j)*`if`(i>n, x, 1)), j=1..i))
    end:
T:= n-> (p-> seq(coeff(p, x, n-i), i=0..n))(b((n+1)$2)):
seq(T(n), n=0..10);  # Alois P. Heinz, Nov 28 2015

t[n_, k_] := Sum[ CatalanNumber[n - j]*CatalanNumber[j], {j, 0, k}]; Flatten[ Table[t[n, k], {n, 0, 9}, {k, n, 0, -1}]] (* Jean-François Alcover, Jul 17 2013 *)

a(n,m) = A028364(n,n-m), n>=m>=0, else 0.
G.f. for column m>=1 (without leading zeros): (c(x)^3)sum(C(m-1, k)*c(x)^k, k=0..m-1), with C(n, m) := (m+1)*binomial(2*n-m, n-m)/(n+1) (Catalan convolutions A033184); and for m=0: c^2(x), where c(x) is g.f. of A000108 (Catalan).
T(n,k) = Sum_{j>=0} A039598(n-k,j)*A039599(k,j). - Philippe Deléham, Feb 18 2004
G.f. for diagonal sequences: see g.f. for columns of A028364.

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

Original entry on oeis.org

1, 8, 84, 1008, 13090, 179088, 2542512, 37106784, 553270671, 8391423040, 129058047580, 2008018827360, 31550226597162, 499892684834368, 7978140653296800, 128138773298754240, 2069603881026760323, 33593111381834512200, 547698081896206040800, 8965330544164089648000, 147285313888568167177866

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 7, r = 8.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A002296, A233832, A233833, A143547, A233834, A233835, A233907, A233908.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[8*Binomial(7*n+8, n)/(7*n+8): n in [0..30]];

Table[8 Binomial[7 n + 8, n]/(7 n + 8), {n, 0, 30}]

PARI
```
a(n) = 8*binomial(7*n+8,n)/(7*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 8.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^8), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/8) is the o.g.f. for A002296. (End)

A081290 a(0) = 0, and for n >=1, a(n) = the largest Catalan number <= n.

Original entry on oeis.org

0, 1, 2, 2, 2, 5, 5, 5, 5, 5, 5, 5, 5, 5, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42

Offset: 0

Antti Karttunen, Mar 17 2003

nonn

After n>0, A000108(n) occurs A000245(n) times.
For all n>0, a(A000108(n)) = A000108(n) [the first occurrence of the n-th Catalan number in this sequence].
Minimal i such that A081289(i) >= A081289(n) [the original definition of the sequence].
In other words, the first position k in A081289 where A081289(n) occurs (the minimal k such that A081289(k) = A081289(n)), and also the first position k in A081288 where A081288(n) occurs (the minimal k such that A081288(k) = A081288(n)). The starting point of the run which contains the n-th term in those sequences.

Cf. A000108, A081288, A081289, A072643, A239903.

Join[{0},With[{catnos=Reverse[CatalanNumber[Range[10]]]},Table[ SelectFirst[ catnos,#<=n&],{n,80}]]] (* This program uses the SelectFirst function from Mathematica version 10 *) (* Harvey P. Dale, Jul 27 2014 *)

a(0) = 0, a(n) = A000108(A081288(n)-1).

Sum_{n>=1} 1/a(n)^2 = 44*Pi/sqrt(3) - 4*Pi^2 - 38. - Amiram Eldar, Aug 18 2022

Name changed by Antti Karttunen, Apr 26 2014

A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).

Original entry on oeis.org

1, 6, 45, 380, 3450, 32886, 324632, 3290040, 34034715, 357919100, 3815041230, 41124015036, 447534498320, 4910258796240, 54257308779600, 603260892430960, 6744185681876505, 75764901779438850, 854867886710698755, 9683529727259434200

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 5, r = 6.

C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
J-C. Aval, Multivariate Fuss-Catalan Numbers, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number

Cf. A000108, A002294, A118969, A143546, A118971, A233669, A233736, A233737, A233738.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[6*Binomial(5*n+6,n)/(5*n+6): n in [0..30]];

Table[6 Binomial[5 n + 6, n]/(5 n + 6), {n, 0, 30}]

PARI
```
a(n) = 6*binomial(5*n+6,n)/(5*n+6);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/6))^6+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, here p = 5, r = 6.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^6), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/6) is the o.g.f. for A002294. (End)
D-finite with recurrence 8*n*(4*n+5)*(2*n+3)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A244160 a(0)=0, and for n >= 1, a(n) = the largest k such that k-th Catalan number <= n.

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6

Offset: 0

Antti Karttunen, Jun 23 2014

nonn

Apart from 0, each n occurs A000245(n) times.

For n >= 1, a(n) gives the largest k such that C(k) <= n, where C(k) stands for the k-th Catalan number, A000108(k).

			For n=1, the largest k such that C(k) <= 1 is 1, thus a(1) = 1.
For n=2, the largest k such that C(k) <= 2 is 2, thus a(2) = 2.
For n=3, the largest k such that C(k) <= 3 is 2, thus a(3) = 2.
For n=4, the largest k such that C(k) <= 4 is 2, thus a(4) = 2.
For n=5, the largest k such that C(k) <= 5 is 3, thus a(5) = 3.

Michael De Vlieger, Table of n, a(n) for n = 0..4861

After zero, one less than A081288.

Cf. A000108, A000245, A081290, A014418, A239903, A244215, A244159, A236859, A126307.

MapIndexed[ConstantArray[First@ #2 - 1, #1] &, Differences@ Array[CatalanNumber, 8, 0]] /. {} -> {0} // Flatten (* Michael De Vlieger, Jun 08 2017 *)
Join[{0},Table[PadRight[{},CatalanNumber[n+1]-CatalanNumber[n],n],{n,6}]// Flatten] (* Harvey P. Dale, Aug 23 2021 *)

from sympy import catalan
def a(n):
    if n==0: return 0
    i=1
    while True:
        if catalan(i)>n: break
        else: i+=1
    return i - 1
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 08 2017

(define (A244160 n) (if (zero? n) n (- (A081288 n) 1)))

a(0) = 0, and for n>=1, a(n) = A081288(n)-1.

For all n>=1, A000108(a(n)) = A081290(n).

A325187 Number of integer partitions of n such that the upper-left square of the Young diagram has strictly greater graph-distance from the lower-right boundary than any other square.

Original entry on oeis.org

1, 0, 1, 3, 3, 5, 9, 14, 20, 26, 38, 53, 75, 101, 132, 175, 229, 301, 394, 509, 650, 826, 1043, 1315, 1656, 2074, 2590, 3218, 3975, 4896, 6008, 7361, 8989, 10960, 13323, 16159, 19531, 23553, 28323, 34002, 40723, 48694, 58115, 69249, 82350, 97766, 115832

Offset: 1

Gus Wiseman, Apr 11 2019

nonn

The k-th part of the origin-to-boundary partition of a Young diagram is the number of squares graph-distance k from the lower-right boundary. The sequence gives the number of integer partitions of n whose Young diagram has last part of its origin-to-boundary partition equal to 1.

The Heinz numbers of these partitions are given by A325185.

			The a(1) = 1 through a(8) = 14 partitions:
  (1)  (21)  (22)   (41)    (51)     (61)      (71)
             (31)   (311)   (321)    (322)     (332)
             (211)  (2111)  (411)    (331)     (422)
                            (3111)   (421)     (431)
                            (21111)  (511)     (521)
                                     (3211)    (611)
                                     (4111)    (3221)
                                     (31111)   (3311)
                                     (211111)  (4211)
                                               (5111)
                                               (32111)
                                               (41111)
                                               (311111)
                                               (2111111)

Eric Weisstein's World of Mathematics, Graph Distance.

Cf. A000245, A188674, A325165, A325169, A325183, A325184, A325185, A325187, A325190, A325191.

otb[ptn_]:=Min@@MapIndexed[#1+#2[[1]]-1&,Append[ptn,0]];
Table[Length[Select[IntegerPartitions[n],otb[#]>otb[Rest[#]]&&otb[#]>otb[DeleteCases[#-1,0]]&]],{n,30}]

cp's OEIS Frontend

A229963 a(n) = 11*binomial(10*n + 11, n)/(10*n + 11) .

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A232265 a(n) = 10*binomial(9*n + 10, n)/(9*n + 10).

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A239903 List of Restricted-Growth Strings a_{k-1}a_{k-2}...a_{2}a_{1}, with k=2 and a_1 in {0,1} or k>2, a_{k-1}=1 and a_{j+1}>=1+a_j, for k-1>j>0.

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Julia

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A026012 Second differences of Catalan numbers A000108.

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A067323 Catalan triangle A028364 with row reversion.

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A233835 a(n) = 8*binomial(7*n + 8, n)/(7*n + 8).

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A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

A232265 a(n) = 10binomial(9n + 10, n)/(9*n + 10).

A233835 a(n) = 8binomial(7n + 8, n)/(7*n + 8).

A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).