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.

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A330965 Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 10, 20, 14, 1, 5, 14, 35, 70, 42, 1, 6, 18, 50, 126, 252, 132, 1, 7, 22, 65, 182, 462, 924, 429, 1, 8, 26, 80, 238, 672, 1716, 3432, 1430, 1, 9, 30, 95, 294, 882, 2508, 6435, 12870, 4862, 1, 10, 34, 110, 350, 1092, 3300, 9438, 24310, 48620, 16796
Offset: 0

Views

Author

Andrew Howroyd, Jan 04 2020

Keywords

Examples

			Array begins:
====================================================
n\k |   0    1    2    3     4     5     6     7
----+-----------------------------------------------
  0 |   1    1    1    1     1     1     1     1 ...
  1 |   1    2    3    4     5     6     7     8 ...
  2 |   2    6   10   14    18    22    26    30 ...
  3 |   5   20   35   50    65    80    95   110 ...
  4 |  14   70  126  182   238   294   350   406 ...
  5 |  42  252  462  672   882  1092  1302  1512 ...
  6 | 132  924 1716 2508  3300  4092  4884  5676 ...
  7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
  ...

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.

Programs

  • Mathematica
    A330965[n_, k_] := CatalanNumber[n]*(k*n + 1);
Table[A330965[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 24 2025 *)
  • PARI
    T(n, k)={(1 + k*n)*binomial(2*n,n)/(n+1)}

Formula

A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).

A253072 The subsequence A253071(2^n-1).

Original entry on oeis.org

1, 7, 21, 95, 333, 1319, 4837, 18447, 68733, 259447, 972565, 3661535, 13756333, 51754567, 194586181, 731919279, 2752461533, 10352254743, 38932913525, 146424889471, 550683608589, 2071066796007, 7789015542949, 29293584500047, 110169505843517, 414334209685687
Offset: 0

Views

Author

N. J. A. Sloane, Jan 31 2015

Keywords

Comments

A253071 is the Run Length Transform of this sequence.
A253072(2^k-1) = A050476(2^k-1), 0<=k<=3. This is just a coincidence, since it fails at m=4. - Omar E. Pol, Feb 01 2015; N. J. A. Sloane, Feb 20 2015

Links

Crossrefs

Cf. A253067, A253068, A253071, A050476.

Programs

  • Maple
    OddCA2:=proc(f,M) local n,a,i,f2,g,p;
f2:=simplify(expand(f)) mod 2;
p:=1; g:=f2;
for n from 1 to M do p:=expand(p*g) mod 2; print(n,nops(p)); g:=expand(g^2) mod 2; od:
return;
end;
f25:=1/(x*y)+1/x+1/y+y+x/y+x+x*y;
OddCA2(f25,8);
  • Mathematica
    LinearRecurrence[{6, -5, -24, 44, -8}, {1, 7, 21, 95, 333}, 26] (* Jean-François Alcover, Nov 27 2017 *)
  • PARI
    Vec(-(8*x^4-28*x^3+16*x^2-x-1)/(8*x^5-44*x^4+24*x^3+5*x^2-6*x+1) + O(x^30)) \\ Colin Barker, Jul 16 2015

Formula

G.f.: -(-1-t+16*t^2-28*t^3+8*t^4)/(1-6*t+5*t^2+24*t^3-44*t^4+8*t^5).
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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