A141351 -id:A141351 - cp's OEIS frontend

A208615 Number of Young tableaux A(n,k) with n k-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing); square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 6, 10, 1, 1, 1, 1, 15, 53, 35, 1, 1, 1, 1, 43, 491, 587, 126, 1, 1, 1, 1, 133, 6091, 25187, 7572, 462, 1, 1, 1, 1, 430, 87781, 1676707, 1725819, 109027, 1716, 1, 1, 1, 1, 1431, 1386529, 140422657, 705002611, 144558247, 1705249, 6435, 1, 1

Offset: 0

Alois P. Heinz, Feb 29 2012

A(n,k) is also the number of (n*k-1)-step walks on k-dimensional cubic lattice from (1,0,...,0) to (n,n,...,n) with positive unit steps in all dimensions such that for each point (p_1,p_2,...,p_k) we have p_1<=p_2<=...<=p_k or p_1>=p_2>=...>=p_k.

			A(2,3) = 6:
  +---+      +---+      +---+      +---+      +---+      +---+
  |123|      |123|      |124|      |125|      |134|      |135|
  |456|      |654|      |356|      |346|      |256|      |246|
  +---+---+  +---+---+  +---+---+  +---+---+  +---+---+  +---+---+
  |x  |100|  |x  |100|  |x  |100|  |x  |100|  |x  |100|  |x  |100|
  | x |110|  | x |110|  | x |110|  | x |110|  |x  |200|  |x  |200|
  |  x|111|  |  x|111|  |x  |210|  |x  |210|  | x |210|  | x |210|
  |x  |211|  |  x|112|  |  x|211|  | x |220|  |  x|211|  | x |220|
  | x |221|  | x |122|  | x |221|  |  x|221|  | x |221|  |  x|221|
  |  x|222|  |x  |222|  |  x|222|  |  x|222|  |  x|222|  |  x|222|
  +---+---+  +---+---+  +---+---+  +---+---+  +---+---+  +---+---+
Square array A(n,k) begins:
  1, 1,   1,      1,         1,            1,                1, ...
  1, 1,   1,      1,         1,            1,                1, ...
  1, 1,   3,      6,        15,           43,              133, ...
  1, 1,  10,     53,       491,         6091,            87781, ...
  1, 1,  35,    587,     25187,      1676707,        140422657, ...
  1, 1, 126,   7572,   1725819,    705002611,     396803649991, ...
  1, 1, 462, 109027, 144558247, 398084427253, 1672481205752413, ...

Alois P. Heinz, Antidiagonals n = 0..25, flattened

Rows 0+1, 2-10 give: A000012, A141351 (for n>1), A208616, A208617, A208618, A208619, A208620, A208621, A208622, A208623.
Columns 0+1, 2-10 give: A000012, A088218, A185148, A208624, A208625, A208626, A208627, A208628, A208629, A208630.
Main diagonal gives: A208631.
Antidiagonal sums give: A208729.

b:= proc() option remember;
      `if`(nargs<2, 1, `if`(args[1]=args[nargs],
      `if`(args[1]=0, 1, 2* b(args[1]-1, seq(args[i], i=2..nargs))),
      `if`(args[1]>0, b(args[1]-1, seq(args[i], i=2..nargs)), 0)
          +add(`if`(args[j]>args[j-1], b(seq(args[i] -`if`(i=j, 1, 0)
                , i=1..nargs)), 0), j=2..nargs) ))
    end:
A:= (n, k)-> `if`(n=0 or k=0, 1, b(n-1, n$(k-1))):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[args__] := b[args] = If[(nargs = Length[{args}]) < 2, 1, If[First[{args}] == Last[{args}], If[First[{args}] == 0, 1, 2*b[First[{args}]-1, Sequence @@ Rest[{args}]]], If[First[{args}] > 0, b[First[{args}]-1, Sequence @@ Rest[{args}]], 0] + Sum [If[{args}[[j]] > {args}[[j-1]], b[Sequence @@ Table[{args}[[i]] - If[i == j, 1, 0], {i, 1, nargs}]], 0], {j, 2, nargs}] ] ]; a[n_, k_] := If[n == 0 || k == 0, 1, b[n-1, Sequence @@ Array[n&, k-1]]]; Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Dec 12 2013, translated from Maple *)

A176137 Number of partitions of n into distinct Catalan numbers, cf. A000108.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Reinhard Zumkeller, Apr 09 2010

nonn

a(n) <= 1;
a(A000108(n)) = 1; a(A141351(n)) = 1; a(A014138(n)) = 1.
A197433 gives all such numbers k that a(k) = 1, in other words, this is the characteristic function of A197433, and all three sequences mentioned above are its subsequences. - Antti Karttunen, Jun 25 2014

			56 = 42+14 = A000108(5)+A000108(4), all other sums of distinct Catalan numbers are not equal 56, therefore a(56)=1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

When right-shifted (prepended with 1) this sequence is the first differences of A244230.

Cf. A033552, A197433, A161227 - A161239.

nmax = 104;
A197433 = CoefficientList[(1/(1 - x))*Sum[ CatalanNumber[k + 1]*x^(2^k)/(1 + x^(2^k)), {k, 0, Log[2, nmax] // Ceiling}] + O[x]^nmax, x];
a[n_] := Boole[MemberQ[A197433, n]];
Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 18 2021, after Ilya Gutkovskiy in A197433 *)

(define (A176137 n) (if (zero? n) 1 (- (A244230 (+ n 1)) (A244230 n)))) ;; Antti Karttunen, Jun 25 2014

a(n) = f(n,1,1) with f(m,k,c) = if c>m then 0^m else f(m-c,k+1,c') + f(m,k+1,c') where c'=2*c*(2*k+1)/(k+2).

A230061 Primes of the form Catalan(n)+1.

Original entry on oeis.org

2, 3, 43, 58787, 4861946401453, 337485502510215975556783793455058624701, 4180080073556524734514695828170907458428751314321, 1000134600800354781929399250536541864362461089950801, 944973797977428207852605870454939596837230758234904051

Offset: 1

K. D. Bajpai, Oct 08 2013

nonn

The 25th term a(25) in the sequence has 693 digits.

a(26) has 1335 digits; a(27) has 1647 digits; a(28) has 1694 digits; a(29) has 2554 digits; a(30) has 4857 digits; a(31) has 4876 digits; a(32) has 9641 digits. - Charles R Greathouse IV, Oct 09 2013

			a(3)= 43: Catalan(5)= (2*5)!/(5!*(5+1)!)= 42. Catalan(5)+1= 43 which is prime.
a(4)= 58787: Catalan(11)= (2*11)!/(11!*(11+1)!)= 58786. Catalan(11)+1= 58787 which is prime.

K. D. Bajpai, Table of n, a(n) for n = 1..25
Chris K. Caldwell and G. L. Honaker,Jr., Prime Curios! 4250

Cf. A000108, A024492, A141351.

Cf. A053429 (numbers n such that Catalan(n)+1 is prime).

KD:= proc() local a,b,c; a:= (2*n)!/(n!*(n + 1)!); b:=a+1;if isprime(b) then return(b): fi; end: seq(KD(),n=1..50);

Select[CatalanNumber[Range[100]]+1,PrimeQ] (* Harvey P. Dale, Aug 26 2021 *)

for(n=1,1e3,if(ispseudoprime(t=binomial(2*n,n)/(n+1)+1),print1(t", "))) \\ Charles R Greathouse IV, Oct 08 2013

A141352 Expansion of (1-2x-5x^2-7x^3+x^6)/((1-x)(1-x^3)^2).

Original entry on oeis.org

1, -1, -6, -11, -15, -25, -35, -41, -56, -71, -79, -99, -119, -129, -154, -179, -191, -221, -251, -265, -300, -335, -351, -391, -431, -449, -494, -539, -559, -609, -659, -681, -736, -791, -815, -875, -935, -961, -1026, -1091, -1119

Offset: 0

Paul Barry, Jun 27 2008

Hankel transform of A141351.

Index entries for linear recurrences with constant coefficients, signature (1,0,2,-2,0,-1,1).

CoefficientList[Series[(1-2x-5x^2-7x^3+x^6)/ ((1-x)(1-x^3)^2), {x,0,50}],x] (* Harvey P. Dale, Mar 31 2011 *)

G.f.: (1-2x-5x^2-7x^3+x^6)/(1-x-2x^3+2x^4+x^6-x^7);

A141353 a(n) = Catalan(n) + 2^n - 0^n.

Original entry on oeis.org

1, 3, 6, 13, 30, 74, 196, 557, 1686, 5374, 17820, 60834, 212108, 751092, 2690824, 9727613, 35423206, 129775862, 477900844, 1767787478, 6565168996, 24468364172, 91486757944, 343068002258, 1289920924540, 4861979955884

Offset: 0

Paul Barry, Jun 27 2008

Hankel transform is A141354.

Cf. A000108 (Catalan numbers), A141351.

f[n_] := Binomial[2n, n]/(n + 1) + 2^n - 0^n; f[0] = 1; Array[f, 29, 0] (* or *)
CoefficientList[ Series[1 + 1/2 (-4 + 2/(1 - 2x) + (1 - Sqrt[1 - 4x])/x), {x, 0, 28}], x] (* Robert G. Wilson v, Mar 18 2018 *)

a(n) = binomial(2*n,n)/(n+1) + 2^n - 0^n; \\ Michel Marcus, Mar 18 2018

G.f.: c(x)+2x/(1-2x), where c(x) is the g.f. of A000108. [corrected by Paul Barry, Oct 18 2010]

Conjecture: (n+1)*a(n) + 2*(-4*n+1)*a(n-1) + 4*(5*n-7)*a(n-2) + 8*(-2*n+5)*a(n-3) = 0. - R. J. Mathar, Nov 15 2012

A141364 a(n) = C(n)-1+0^n where C(n) = A000108(n).

Original entry on oeis.org

1, 0, 1, 4, 13, 41, 131, 428, 1429, 4861, 16795, 58785, 208011, 742899, 2674439, 9694844, 35357669, 129644789, 477638699, 1767263189, 6564120419, 24466267019, 91482563639, 343059613649, 1289904147323, 4861946401451, 18367353072151

Offset: 0

Paul Barry, Jun 27 2008

Hankel transform is A141365.

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

Cf. A000108, A001453 (essentially the same sequence), A141351.

CoefficientList[Series[(1 - Sqrt[1 - 4*x]) / (2*x)-x/(1-x),{x,0,26}],x] (* James C. McMahon, Jul 23 2025 *)

G.f.: c(x)-x/(1-x) where c(x) is the g.f. of A000108

A352507 Number whose representation in the base of Catalan numbers (A014418) is palindromic.

Original entry on oeis.org

0, 1, 3, 6, 8, 15, 22, 43, 48, 53, 59, 64, 69, 133, 152, 171, 177, 196, 215, 430, 444, 458, 477, 491, 505, 524, 538, 552, 564, 578, 592, 611, 625, 639, 658, 672, 686, 1431, 1487, 1543, 1568, 1624, 1680, 1705, 1761, 1817, 1862, 1918, 1974, 1999, 2055, 2111, 2136

Offset: 1

Amiram Eldar, Mar 19 2022

The partial sums of the Catalan numbers with positive index (A014138) are terms, since the representation of A014138(n) is n 1's.

			The first 10 terms are:
   n  a(n)  A014418(a(n))
  --  ----  -------------
   1     0              0
   2     1              1
   3     3             11
   4     6            101
   5     8            111
   6    15           1001
   7    22           1111
   8    43          10001
   9    48          10101
  10    53          10201

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000108, A014418.
Subsequences: A014138, A141351 \ {2}.
Similar sequences: A002113, A006995, A014190, A094202, A331191, A351712, A351717, A352087, A352105, A352319, A352341.

c[n_] := c[n] = CatalanNumber[n]; q[n_] := Module[{s = {}, m = n, i}, While[m > 0, i = 1; While[c[i] <= m, i++]; i--; m -= c[i]; AppendTo[s, i]]; PalindromeQ @ IntegerDigits[Total[4^(s - 1)], 4]]; Select[Range[0, 2000], q]

cp's OEIS Frontend

A208615 Number of Young tableaux A(n,k) with n k-length rows, increasing entries down the columns and monotonic entries along the rows (first row increasing); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A176137 Number of partitions of n into distinct Catalan numbers, cf. A000108.

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A230061 Primes of the form Catalan(n)+1.

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A141352 Expansion of (1-2x-5x^2-7x^3+x^6)/((1-x)(1-x^3)^2).

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A141353 a(n) = Catalan(n) + 2^n - 0^n.

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A141364 a(n) = C(n)-1+0^n where C(n) = A000108(n).

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A352507 Number whose representation in the base of Catalan numbers (A014418) is palindromic.

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