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A176282 Triangle T(n,k) = 1 + A000330(n) - A000330(k) - A000330(n-k), read by rows. 2

%I

%S 1,1,1,1,4,1,1,9,9,1,1,16,21,16,1,1,25,37,37,25,1,1,36,57,64,57,36,1,

%T 1,49,81,97,97,81,49,1,1,64,109,136,145,136,109,64,1,1,81,141,181,201,

%U 201,181,141,81,1,1,100,177,232,265,276,265,232,177,100,1

%N Triangle T(n,k) = 1 + A000330(n) - A000330(k) - A000330(n-k), read by rows.

%C Not summing squares but summing integers implied by the definition (i.e., not using A000330 but A000217) gives A077028.

%C Row sums = {1, 2, 6, 20, 55, 126, 252, 456, 765, 1210, 1826, ...} = (n+1)*(n+2)*(n^2-2*n+3)/6.

%H G. C. Greubel, <a href="/A176282/b176282.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = T(n,n-k).

%F T(n,k) = 1 + k*(n+1)*(n-k). - _G. C. Greubel_, Nov 24 2019

%e Triangle begins as:

%e 1;

%e 1, 1;

%e 1, 4, 1;

%e 1, 9, 9, 1;

%e 1, 16, 21, 16, 1;

%e 1, 25, 37, 37, 25, 1;

%e 1, 36, 57, 64, 57, 36, 1;

%e 1, 49, 81, 97, 97, 81, 49, 1;

%e 1, 64, 109, 136, 145, 136, 109, 64, 1;

%e 1, 81, 141, 181, 201, 201, 181, 141, 81, 1;

%e 1, 100, 177, 232, 265, 276, 265, 232, 177, 100, 1;

%p seq(seq(1 + k*(n+1)*(n-k), k=0..n), n=0..12); # _G. C. Greubel_, Nov 24 2019

%t (* Sequence for q=1..10 *)

%t f[n_, k_, q_]:= f[n, k, q] = 1 +Sum[i^q, {i,0,n}] -Sum[i^q, {i,0,k}] + Sum[i^q, {i,0,n-k}]; Table[Flatten[Table[f[n, k, q],{n,0,12}, {k,0,n}]], {q,1,10}]

%t (* Second program *)

%t Table[1 + k*(n+1)*(n-k), {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 24 2019 *)

%o (PARI) T(n,k) = 1 + k*(n+1)*(n-k); \\ _G. C. Greubel_, Nov 24 2019

%o (MAGMA) [1 + k*(n+1)*(n-k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 24 2019

%o (Sage) [[1 + k*(n+1)*(n-k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 24 2019

%o (GAP) Flat(List([0..12], n-> List([0..n], k-> 1 + k*(n+1)*(n-k) ))); # _G. C. Greubel_, Nov 24 2019

%Y Cf. A077028.

%K nonn,tabl,easy

%O 0,5

%A _Roger L. Bagula_, Apr 14 2010

%E Edited by _R. J. Mathar_, May 03 2013

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Last modified July 4 14:51 EDT 2020. Contains 335448 sequences. (Running on oeis4.)