A143219 Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

1, 2, 6, 3, 9, 18, 4, 12, 24, 40, 5, 15, 30, 50, 75, 6, 18, 36, 60, 90, 126, 7, 21, 42, 70, 105, 147, 196, 8, 24, 48, 80, 120, 168, 224, 288, 9, 27, 54, 90, 135, 189, 252, 324, 405, 10, 30, 60, 100, 150, 210, 280, 360, 450, 550

Offset

1,2

Links

G. C. Greubel, Rows n = 1..50 of the triangle, flattened

Formula

Triangle read by rows, A127648 * A000012 * A127773, 1 <= k <= n.

Sum_{k=1..n} T(n, k) = A002417(n).

T(n, n) = A002411(n).

From G. C. Greubel, Jul 12 2022: (Start)

T(n, k) = A002024(n,k) * A127773(n,k).

T(n, k) = n * binomial(k+1, 2).

Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/4)*(4*n - 3*floor((n+1)/2) + 3)*binomial(2 + floor((n+1)/2), 3).

T(2*n-1, n) = A002414(n), n >= 1.

T(2*n-2, n-1) = A011379(n-1), n >= 2. (End)

Example

First few rows of the triangle =
  1;
  2,  6;
  3,  9, 18;
  4, 12, 24, 40;
  5, 15, 30, 50,  75;
  6, 18, 36, 60,  90, 126;
  7, 21, 42, 70, 105, 147, 196;
  ...

Mathematica

Table[n*Binomial[k+1, 2], {n, 12}, {k, n}]//Flatten (* G. C. Greubel, Jul 12 2022 *)

Prog

(Magma) [n*Binomial(k+1, 2): k in [1..n], n in [1..12]]; // G. C. Greubel, Jul 12 2022
(SageMath) flatten([[n*binomial(k+1, 2) for k in (1..n)] for n in (1..12)]) # G. C. Greubel, Jul 12 2022

Crossrefs

Cf. A000012, A127648, A127773.

Cf. A002024, A002411 (right border), A002414, A002417 (row sums), A011379.

Sequence in context: A378822 A079297 A276941 * A109465 A090705 A208547

Adjacent sequences: A143216 A143217 A143218 * A143220 A143221 A143222

Keyword

nonn,tabl

Author

Gary W. Adamson, Jul 30 2008

Status

approved