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A323231 A(n, k) = [x^k] (1/(1-x) + x/(1-x)^n), square array read by descending antidiagonals for n, k >= 0. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

REFERENCES

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

LINKS

Table of n, a(n) for n=0..77.

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).

EXAMPLE

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.

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);

PROG

(Sage)

def Arow(n):

    R.<x> = PowerSeriesRing(ZZ, 20)

    gf = 1/(1-x) + x/(1-x)^n

    return gf.padded_list(10)

for n in (0..9): print(Arow(n))

(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

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).

Sequence in context: A220424 A182907 A334745 * A175128 A205154 A337772

Adjacent sequences:  A323228 A323229 A323230 * A323232 A323233 A323234

KEYWORD

nonn,tabl

AUTHOR

Peter Luschny, Feb 10 2019

STATUS

approved

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Last modified June 16 14:16 EDT 2021. Contains 345057 sequences. (Running on oeis4.)