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A094415 Triangle T read by rows: dot product <r,r-1,...,1> * <s+1,s+2,...,r,1,2,...,s>. 10
1, 4, 5, 10, 13, 13, 20, 26, 28, 26, 35, 45, 50, 50, 45, 56, 71, 80, 83, 80, 71, 84, 105, 119, 126, 126, 119, 105, 120, 148, 168, 180, 184, 180, 168, 148, 165, 201, 228, 246, 255, 255, 246, 228, 201, 220, 265, 300, 325, 340, 345, 340, 325, 300, 265, 286, 341 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

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

FORMULA

T(n, k) = n*(n^2 + 3*n*(1+k) + 2 - 3*k^2)/6 for n >= 0, 0 <= k <= n.

EXAMPLE

Triangle begins as:

   1;

   4,  5;

  10, 13, 13;

  20, 26, 28, 26;

  35, 45, 50, 50, 45;

  56, 71, 80, 83, 80, 71;

MAPLE

seq(seq( (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 , k=0..n), n=0..12); # G. C. Greubel, Oct 30 2019

MATHEMATICA

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

PROG

(PARI) T(n, k) = (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6;

for(n=0, 12, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Oct 30 2019

(MAGMA) [(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6: k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 30 2019

(Sage) [[(n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 for k in (0..n)] for n in (0..12)] # G. C. Greubel, Oct 30 2019

(GAP) Flat(List([0..12], n-> List([0..n], k-> (n+1)*((n+2)*(n+3) + 3*k*(n-k+1))/6 ))); # G. C. Greubel, Oct 30 2019

CROSSREFS

Columns 0-6 are A000292, A008778, A026054, A026057, A026060, A026063, A026066.

Half-diagonal is A050410.

Row sums are A000537.

See also A094414, A088003.

Sequence in context: A217571 A058335 A222353 * A114517 A283246 A236283

Adjacent sequences:  A094412 A094413 A094414 * A094416 A094417 A094418

KEYWORD

nonn,tabl

AUTHOR

Ralf Stephan, May 02 2004

STATUS

approved

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Last modified November 16 17:03 EST 2019. Contains 329201 sequences. (Running on oeis4.)