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)).

A172171 (1, 9) Pascal Triangle read by horizontal rows. Same as A093644, but mirrored and without the additional row/column (1, 9, 9, 9, 9, ...).

Original entry on oeis.org

1, 1, 10, 1, 11, 19, 1, 12, 30, 28, 1, 13, 42, 58, 37, 1, 14, 55, 100, 95, 46, 1, 15, 69, 155, 195, 141, 55, 1, 16, 84, 224, 350, 336, 196, 64, 1, 17, 100, 308, 574, 686, 532, 260, 73, 1, 18, 117, 408, 882, 1260, 1218, 792, 333, 82
Offset: 1

Views

Author

Mark Dols, Jan 28 2010

Keywords

Comments

Binomial transform of A017173.

Examples

			Triangle begins:
  1;
  1, 10;
  1, 11,  19;
  1, 12,  30,  28;
  1, 13,  42,  58,   37;
  1, 14,  55, 100,   95,   46;
  1, 15,  69, 155,  195,  141,   55;
  1, 16,  84, 224,  350,  336,  196,   64;
  1, 17, 100, 308,  574,  686,  532,  260,   73;
  1, 18, 117, 408,  882, 1260, 1218,  792,  333,   82;
  1, 19, 135, 525, 1290, 2142, 2478, 2010, 1125,  415,  91;
  1, 20, 154, 660, 1815, 3432, 4620, 4488, 3135, 1540, 506, 100;

Links

Crossrefs

Cf. A007318, A017173, A050489 (central terms), A093644, A139634 (row sums).

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[k<1 || k>n, 0, If[k==1, 1, If[n==2 && k==2, 10, T[n-1, k] + 2*T[n-1, k-1] - T[n-2, k-1] - T[n-2, k-2]]]];
Table[T[n, k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Apr 24 2022 *)
  • SageMath
    @CachedFunction
def T(n,k):
    if (k<1 or k>n): return 0
    elif (k==1): return 1
    elif (n==2 and k==2): return 10
    else: return T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2)
flatten([[T(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 24 2022

Formula

T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) - T(n-2,k-2), T(n,1) = 1, T(2,2) = 10, T(n,k) = 0 if k < 1 or if k > n.
Sum_{k=0..n} T(n, k) = A139634(n).
T(2*n-1, n) = A050489(n).

Extensions

More terms from Philippe Deléham, Dec 25 2013
Showing 1-2 of 2 results.

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

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