A094414 Triangle T read by rows: dot product <1,2,...,r> * <s+1,s+2,...,r,1,2,...,s>.

1, 5, 4, 14, 11, 11, 30, 24, 22, 24, 55, 45, 40, 40, 45, 91, 76, 67, 64, 67, 76, 140, 119, 105, 98, 98, 105, 119, 204, 176, 156, 144, 140, 144, 156, 176, 285, 249, 222, 204, 195, 195, 204, 222, 249, 385, 340, 305, 280, 265, 260, 265, 280, 305, 340, 506, 451, 407, 374, 352, 341, 341, 352, 374, 407, 451

Offset

0,2

Comments

Offset for r (the rows) is 1, for s (the columns) it is 0.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

T(r, s) = r*(2*r^2 + 3*r - 3*r*s + 1 + 3*s^2)/6, r >= 1, 0 <= s <= r-1.

Example

Triangle begins as:
   1;
   5,  4;
  14, 11, 11;
  30, 24, 22, 24;
  55, 45, 40, 40, 45;
  91, 76, 67, 64, 67, 76;

Maple

T:=proc(r, s) if s>=r then 0 else r*(2*r^2+3*r+1-3*r*s+3*s^2)/6 fi end: for r from 1 to 11 do seq(T(r, s), s=0..r-1) od; # yields sequence in triangular form # Emeric Deutsch, Nov 27 2006

Mathematica

Table[n*((n+1)*(2*n+1) -3*k*(n-k))/6, {n, 0, 12}, {k, 0, n-1}]//Flatten (* G. C. Greubel, Oct 30 2019 *)

Prog

(PARI) T(n, k) = n*((n+1)*(2*n+1) -3*k*(n-k))/6;
for(n=0, 12, for(k=0, n-1, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 30 2019
(Magma) [n*((n+1)*(2*n+1) -3*k*(n-k))/6: k in [0..n-1], n in [0..12]]; // G. C. Greubel, Oct 30 2019
(Sage) [[n*((n+1)*(2*n+1) -3*k*(n-k))/6 for k in (0..n-1)] for n in (0..12)] # G. C. Greubel, Oct 30 2019
(GAP) Flat(List([0..12], n-> List([0..n-1], k-> n*((n+1)*(2*n+1) -3*k*(n-k))/6 ))); # G. C. Greubel, Oct 30 2019

Crossrefs

Columns 0-6 are A000330, A006527, A026037, A026040, A026043, A026046, A026049.

Row sums are A000537.

See also A094415, A088003.

Sequence in context: A078930 A329162 A344817 * A158867 A107984 A133178

Adjacent sequences: A094411 A094412 A094413 * A094415 A094416 A094417

Keyword

nonn,tabl

Author

Ralf Stephan, May 02 2004

Extensions

More terms from G. C. Greubel, Oct 30 2019

Status

approved