A237124 Triangle of numbers related to Catalan numbers (A000108).

1, 1, 1, 1, 2, 1, 3, 4, 3, 1, 9, 11, 8, 4, 1, 28, 33, 24, 13, 5, 1, 90, 104, 76, 43, 19, 6, 1, 297, 339, 249, 145, 69, 26, 7, 1, 1001, 1133, 836, 497, 248, 103, 34, 8, 1, 3432, 3861, 2860, 1727, 891, 394, 146, 43, 9, 1, 11934, 13364, 9932, 6071, 3211, 1484, 593, 199, 53, 10, 1

Offset

0,5

Comments

Riordan array (1 +x +x^2*C(x)^3, x*C(x)) where C(x) is the g.f. of A000108.

Diagonal sums are A000108(n).

Row sums are T(n+1,1).

T(n,0) = A071724(n-1).

T(n,1) = A220902(n), n>=2.

T(n,2) = A228404(n-2), n>=4.

T(n+3,3) = A033434(n).

T(n,n) = 1.

T(n+1,n) = n+1.

T(n+2,n) = A034856(n+1).

Links

G. C. Greubel, Rows n = 0..30 of the triangle, flattened

Formula

From Peter Bala, Feb 18 2018: (Start)

T(n,k) = C(2*n+1-k, n-k) - 2*C(2*n-k, n-k-1) - C(2*n-1-k, n-k-2) + 3*C(2*n-2-k, n-k-3) - 2*C(2*n-3-k, n-k-4), for n > 2, otherwise C(n, k).

The n-th row polynomial of the row reverse triangle equals the n-th degree Taylor polynomial of the function (1 - x^2 + x^3)*(1 - 2*x)/(1 - x)^2 * 1/(1 - x)^n about 0. For example, for n = 4, (1 - x^2 + x^3)*(1 - 2*x)/(1 - x)^2 * 1/(1 - x)^4 = 1 + 4*x + 8*x^2 + 11*x^3 + 9*x^4 + O(x^5), giving (9, 11, 8, 4, 1) as row 4. (End)

Example

Triangle begins:
      1;
      1,     1;
      1,     2,    1;
      3,     4,    3,    1;
      9,    11,    8,    4,    1;
     28,    33,   24,   13,    5,    1;
     90,   104,   76,   43,   19,    6,   1;
    297,   339,  249,  145,   69,   26,   7,   1;
   1001,  1133,  836,  497,  248,  103,  34,   8,  1;
   3432,  3861, 2860, 1727,  891,  394, 146,  43,  9,  1;
  11934, 13364, 9932, 6071, 3211, 1484, 593, 199, 53, 10, 1;
  ...

Mathematica

b[n_, k_]:= Binomial[2*n-k+1, n-k];
T[n_, k_]:= If[n<3, Binomial[n, k], b[n, k] -2*b[n, k+1] -b[n, k+2] +3*b[n, k+3] - 2*b[n, k+4]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 08 2021 *)

Prog

(Sage)
def b(n, k): return binomial(2*n-k+1, n-k)
def T(n, k): return binomial(n, k) if (n<3) else b(n, k) -2*b(n, k+1) -b(n, k+2) +3*b(n, k+3) -2*b(n, k+4)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021

Crossrefs

Cf. A000108.

Sequence in context: A367019 A193020 A301471 * A233547 A358193 A122530

Adjacent sequences: A237121 A237122 A237123 * A237125 A237126 A237127

Keyword

nonn,tabl,easy

Author

Philippe Deléham, Feb 03 2014

Status

approved