A323231 A(n, k) = [x^k] (1/(1-x) + x/(1-x)^n), square array read by descending antidiagonals for n, k >= 0.

1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 1, 2, 4, 4, 2, 1, 1, 2, 5, 7, 5, 2, 1, 1, 2, 6, 11, 11, 6, 2, 1, 1, 2, 7, 16, 21, 16, 7, 2, 1, 1, 2, 8, 22, 36, 36, 22, 8, 2, 1, 1, 2, 9, 29, 57, 71, 57, 29, 9, 2, 1, 1, 2, 10, 37, 85, 127, 127, 85, 37, 10, 2, 1

Offset: 0

Author

Peter Luschny, Feb 10 2019

Keywords

• nonn
• tabl

Examples

			Array starts:
[0] 1, 2,  1,  1,   1,   1,    1,    1,    1,     1,     1, ...
[1] 1, 2,  2,  2,   2,   2,    2,    2,    2,     2,     2, ... A040000
[2] 1, 2,  3,  4,   5,   6,    7,    8,    9,    10,    11, ... A000027
[3] 1, 2,  4,  7,  11,  16,   22,   29,   37,    46,    56, ... A000124
[4] 1, 2,  5, 11,  21,  36,   57,   85,  121,   166,   221, ... A050407
[5] 1, 2,  6, 16,  36,  71,  127,  211,  331,   496,   716, ... A145126
[6] 1, 2,  7, 22,  57, 127,  253,  463,  793,  1288,  2003, ... A323228
[7] 1, 2,  8, 29,  85, 211,  463,  925, 1717,  3004,  5006, ...
[8] 1, 2,  9, 37, 121, 331,  793, 1717, 3433,  6436, 11441, ...
[9] 1, 2, 10, 46, 166, 496, 1288, 3004, 6436, 12871, 24311, ...
.
Read as a triangle (by descending antidiagonals):
                                     1
                                  2,   1
                                1,   2,   1
                             1,   2,   2,   1
                           1,   2,   3,   2,   1
                        1,   2,   4,   4,   2,   1
                      1,   2,   5,   7,   5,   2,  1
                    1,  2,   6,  11,  11,   6,   2,  1
                  1,  2,   7,  16,  21,  16,   7,  2,  1
                1,  2,  8,  22,  36,  36,  22,   8,  2,  1
              1,  2,  9,  29,  57,  71,  57,  29,  9,  2,  1
.
A(0, 1) = C(-1, 0) + 1 = 2 because C(-1, 0) = 1. A(1, 0) = C(-1, -1) + 1 = 1 because C(-1, -1) = 0. Warning: Some computer algebra programs (for example Maple and Mathematica) return C(n, n) = 1 for n < 0. This contradicts the definition given by Graham et al. (see reference). On the other hand this definition preserves symmetry.
		

References

• R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 154.

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened

Crossrefs

Differs from A323211 only in the second term.

Rows include: A040000, A000027, A000124, A050407, A145126, A323228.

Diagonals A(n, n+d): A323230 (d=0), A260878 (d=1), A323229 (d=2).

Antidiagonal sums are A323227(n) if n!=1.

Cf. A007318 (Pascal's triangle).

Programs

Julia

using AbstractAlgebra
function Arow(n, len)
    R, x = PowerSeriesRing(ZZ, len+2, "x")
    gf = inv(1-x) + divexact(x, (1-x)^n)
    [coeff(gf, k) for k in 0:len-1] end
for n in 0:9 println(Arow(n, 11)) end

Maple

Binomial := (n, k) -> `if`(n < 0 and n = k, 0, binomial(n,k)):
A := (n, k) -> Binomial(n + k - 2, k - 1) + 1:
seq(lprint(seq(A(n, k), k=0..10)), n=0..10);

Mathematica

T[n_, k_]:= If[k==0, 1 + Boole[n==1], If[k==n, 1, Binomial[n-2, k-1] + 1]];
Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 27 2021 *)

Sage

def Arow(n):
    R. = PowerSeriesRing(ZZ, 20)
    gf = 1/(1-x) + x/(1-x)^n
    return gf.padded_list(10)
for n in (0..9): print(Arow(n))

Formula

A(n, k) = binomial(n + k - 2, k - 1) + 1. Note that binomial(n, n) = 0 if n < 0.

A(n, k) = A(k, n) with the exception A(1,0) != A(0,1).

A(n, n) = binomial(2*n-2, n-1) + 1 = A323230(n).

From G. C. Greubel, Dec 27 2021: (Start)

T(n, k) = binomial(n-2, k-1) + 1 with T(n, 0) = 1 + [n=1], T(n, n) = 1.

T(2*n, n) = A323230(n).

Sum_{k=0..n} T(n,k) = n + 1 + 2^(n-2) - [n=0]/4 + [n=1]/2. (End)