cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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.

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Feb 29 2012

Keywords

Comments

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.

Examples

			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, ...
		

Crossrefs

Rows 0+1, 2-10 give: A000012, A141351 (for n>1), A208616, A208617, A208618, A208619, A208620, A208621, A208622, A208623.
Main diagonal gives: A208631.
Antidiagonal sums give: A208729.

Programs

  • Maple
    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);
  • Mathematica
    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

Views

Author

Reinhard Zumkeller, Apr 09 2010

Keywords

Comments

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

Examples

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

Crossrefs

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

Programs

Formula

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

Views

Author

K. D. Bajpai, Oct 08 2013

Keywords

Comments

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

Examples

			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.
		

Crossrefs

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

Programs

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

Views

Author

Paul Barry, Jun 27 2008

Keywords

Comments

Hankel transform of A141351.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Paul Barry, Jun 27 2008

Keywords

Comments

Hankel transform is A141354.

Crossrefs

Cf. A000108 (Catalan numbers), A141351.

Programs

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

Formula

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

Views

Author

Paul Barry, Jun 27 2008

Keywords

Comments

Hankel transform is A141365.

Crossrefs

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

Programs

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

Formula

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

Views

Author

Amiram Eldar, Mar 19 2022

Keywords

Comments

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

Examples

			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
		

Crossrefs

Subsequences: A014138, A141351 \ {2}.

Programs

  • Mathematica
    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]
Showing 1-7 of 7 results.