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A237124
Triangle of numbers related to Catalan numbers (A000108).
1
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).
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
KEYWORD
nonn,tabl,easy
AUTHOR
Philippe Deléham, Feb 03 2014
STATUS
approved