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

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

Author

Ralf Stephan, May 02 2004

Keywords

• nonn
• tabl

Comments

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

Examples

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

Links

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

Crossrefs

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

Row sums are A000537.

See also A094415, A088003.

Programs

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

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

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 *)

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

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

Formula

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

Extensions

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