A101493 Triangle read by rows: T(n,k) = (n+1)*(2*(n+1)-1) - k*(2*k-1).

1, 6, 5, 15, 14, 9, 28, 27, 22, 13, 45, 44, 39, 30, 17, 66, 65, 60, 51, 38, 21, 91, 90, 85, 76, 63, 46, 25, 120, 119, 114, 105, 92, 75, 54, 29, 153, 152, 147, 138, 125, 108, 87, 62, 33, 190, 189, 184, 175, 162, 145, 124, 99, 70, 37, 231, 230, 225, 216, 203, 186, 165, 140, 111, 78, 41

Offset

0,2

Comments

The triangle is generated from the product B*A of the infinite lower triangular matrices A =

1 0 0 0 ...

1 1 0 0 ...

1 1 1 0 ...

1 1 1 1 ...

... and B =

1 0 0 0 ...

1 5 0 0 ...

1 5 9 0 ...

1 5 9 13 ...

...

Links

Muniru A Asiru, Rows n = 0..150 of triangle, flattened

Formula

T(n,n)*T(n,0) = (n+1)*(2*n+1)*(4*n+1) = A079588(n).

G.f.: (1 - 7*x^2*y + 3*x*(1 + y))/((1 - x)^3*(1 - x*y)^2). - Stefano Spezia, Oct 23 2025

Example

Triangle begins:
   1;
   6,  5;
  15, 14,  9;
  28, 27, 22, 13;
  45, 44, 39, 30, 17;
  66, 65, 60, 51, 38, 21;
  ...

Mathematica

A101493[n_, k_] := (n + 1)*(2*n + 1) - k*(2*k - 1);
Table[A101493[n, k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Oct 24 2025 *)

Prog

(PARI) T(n, k)=if(k>n, 0, (n+1)*(2*(n+1)-1)-k*(2*k-1))
for(i=0, 10, for(j=0, i, print1(T(i, j), ", ")); print())
(GAP) Flat(List([0..10], n->List([0..n], k->(n+1)*(2*n+1)-k*(2*k-1)))); # Muniru A Asiru, Mar 05 2019

Crossrefs

Columns k=0..2 are A000384, A017137, A017557.

Main diagonal is A016813.

Row sums give A007585(n+1).

Cf. A079588, A101492 (for product A*B).

Sequence in context: A123168 A119636 A300750 * A347276 A396283 A188067

Adjacent sequences: A101490 A101491 A101492 * A101494 A101495 A101496

Keyword

nonn,tabl,easy

Author

Lambert Klasen (lambert.klasen(AT)gmx.de) and Gary W. Adamson, Jan 21 2005

Status

approved