A086460 Square array read by antidiagonals: A(n, k) = n*k + 0^k.

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 4, 3, 0, 1, 4, 6, 6, 4, 0, 1, 5, 8, 9, 8, 5, 0, 1, 6, 10, 12, 12, 10, 6, 0, 1, 7, 12, 15, 16, 15, 12, 7, 0, 1, 8, 14, 18, 20, 20, 18, 14, 8, 0, 1, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0, 1, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 0, 1, 11, 20, 27, 32, 35, 36, 35, 32, 27, 20, 11, 0

Offset

0,8

Comments

Inverse binomial transform of array A049513.

The array, and antidiagonal triangle, are symmetric, except for the rows k=0 and n=0 (for the array), and k=0 and k=n (for the triangle). - G. C. Greubel, Jan 21 2026

Links

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

Formula

A(n, k) = n*k + 0^k (array).

T(n, k) = (n-k)*k + 0^k (antidiagonal triangle).

From G. C. Greubel, Jan 21 2026: (Start)

A(n, k) = A(k, n) + [k=0] - [n=0] (symmetric columns and rows for n,k > 0).

G.f.: (1 - x - (2-3*x)*x*y + (1-x)*(x*y)^2)/((1-x)*(1-x*y))^2 (antidiagonal triangle). (End)

Example

Rows of the array, A(n, k), begin as:
  1,  0,  0,  0,  0,  0,  0,  0,  0, ... A000007;  A000004(k) + [k=0];
  1,  1,  2,  3,  4,  5,  6,  7,  8, ... A028310;  A001477(k) + [k=0];
  1,  2,  4,  6,  8, 10, 12, 14, 16, ... A004277;  A005843(k) + [k=0];
  1,  3,  6,  9, 12, 15, 18, 21, 24, ... A008486;  A008585(k) + [k=0];
  1,  4,  8, 12, 16, 20, 24, 28, 32, ... A008574;  A008586(k) + [k=0];
  1,  5, 10, 15, 20, 25, 30, 35, 40, ... A008706;  A008587(k) + [k=0];
  1,  6, 12, 18, 24, 30, 36, 42, 48, ... A008458;  A008588(k) + [k=0];
  1,  7, 14, 21, 28, 35, 42, 49, 56, ... .......;  A008589(k) + [k=0];
  1,  8, 16, 24, 32, 40, 48, 56, 64, ... A022144;  A008590(k) + [k=0];
  1,  9, 18, 27, 36, 45, 54, 63, 72, ... .......;  A008591(k) + [k=0];
  1, 10, 20, 30, 40, 50, 60, 70, 80, ... .......;  A008592(k) + [k=0];
  ...
Rows of the antidiagonals, T(n, k), begin as:
  1;
  1, 0;
  1, 1,  0;
  1, 2,  2,  0;
  1, 3,  4,  3,  0;
  1, 4,  6,  6,  4,  0;
  1, 5,  8,  9,  8,  5,  0;
  1, 6, 10, 12, 12, 10,  6,  0;
  1, 7, 12, 15, 16, 15, 12,  7,  0;
  1, 8, 14, 18, 20, 20, 18, 14,  8, 0;
  1, 9, 16, 21, 24, 25, 24, 21, 16, 9, 0;
  ...

Mathematica

A086460[n_, k_]:= n*k + Boole[k==0];
Table[A086460[n-k, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 21 2026 *)

Prog

(Magma)
A086460:= func< n, k | n*k +0^k >;
[A086460(n-k, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 21 2026
(SageMath)
def A086460(n, k): return n*k +int(k==0)
print(flatten([[A086460(n-k, k) for k in range(n+1)] for n in range(16)])) # G. C. Greubel, Jan 21 2026

Crossrefs

Rows of the array include: A000007, A004277, A008458, A008486, A008574, A008589, A008591, A008592, A008706, A022144, A028310.

Columns of the array include: A000012, A001477, A005843, A008585, A008586, A008587, A008588, A008590.

Sums involving T(n, k) include: A050407 (row), (-1)^(n+1)*A057979(n-1) (signed row), A006918(n-2) + 1 (antidiagonal).

Main diagonal: A000290 (preceded by extra 1).

Diagonals of the form A(n,n+p): A253909 (p=0), A103505 (p=1), A132411 (p=2), A217748 (p=3), A028347 (p=4), A028557 (p=5), A028560 (p=6), A028563 (p=7), A028566 (p=8), A028569 (p=9), A098603 (p=10), A119412 (p=11), A098847 (p=12), A132759 (p=13), A098848 (p=14), A132760 (p=15), A098849 (p=16), A132761 (p=17), A098850 (p=18), A132762 (p=19), A120071 (p=20), A132763 - A132767 (p=21..25).

Diagonals of the form A(2*n,n+p): A001105 (p=0), A046092 (p=1), A054000 (p=2), A139570 (p=3), A067728 (p=4).

Diagonals of the form A(3*n,n+p): A033428 (p=0), A028896 (p=1), A067725 (p=2), A277978 (p=3), A067707 (p=4).

Diagonals of the form A(4*n,n+p): A055808 (p=0), A033996 (p=1), A134582 (p=2), A332519 (p=3).

Diagonals of the form A(5*n,n+p): A033429 (p=0), A124080 (p=1), A067724 (p=2).

Diagonals of the form A(6*n,n+p): A033581 (p=0), A049598 (p=1), A067726 (p=2).

Cf. A000004, A049513.

Sequence in context: A220074 A059259 A124394 * A321884 A136431 A182888

Adjacent sequences: A086457 A086458 A086459 * A086461 A086462 A086463

Keyword

easy,nonn,tabl

Author

Paul Barry, Jul 21 2003

Status

approved