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A055461 Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.

%I A055461 #59 Feb 01 2024 04:19:28
%S A055461 1,4,1,9,4,1,16,9,4,1,25,16,9,4,1,36,25,16,9,4,1,49,36,25,16,9,4,1,64,
%T A055461 49,36,25,16,9,4,1,81,64,49,36,25,16,9,4,1,100,81,64,49,36,25,16,9,4,
%U A055461 1,121,100,81,64,49,36,25,16,9,4,1,144,121,100,81,64,49,36,25,16,9,4,1
%N A055461 Square decrescendo subsequences: triangle T(n,k) = (n-k)^2, n >= 1, 0 <= k < n.
%H A055461 Robert Israel, <a href="/A055461/b055461.txt">Rows n = 1..141, flattened</a>
%F A055461 a(n) = A004736(n)^2.
%F A055461 Sum_{k=0..n-1} T(n, k) = A000330(n) (row sums). - _Michel Marcus_, Dec 31 2012
%F A055461 G.f. as triangle: x*(1+x)/((1-x*y)*(1-x)^3). - _Robert Israel_, Jan 18 2018
%F A055461 Sum_{k=0..n-1} (-1)^k*T(n, k) = A000217(n) (alternating row sums). - _Omar E. Pol_, Jan 24 2014
%F A055461 From _G. C. Greubel_, Jan 31 2024: (Start)
%F A055461 T(2*n-1, n-1) = A000290(n).
%F A055461 Sum_{k=0..floor(n/2)} T(n-k, k) = A000292(n).
%F A055461 Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A194274(n).
%F A055461 Sum_{k=0..floor(n/2)} T(n, k) = A129371(n). (End)
%e A055461 From _Omar E. Pol_, Jan 26 2014: (Start)
%e A055461 Triangle begins:
%e A055461     1;
%e A055461     4,  1;
%e A055461     9,  4,  1;
%e A055461    16,  9,  4,  1;
%e A055461    25, 16,  9,  4,  1;
%e A055461    36, 25, 16,  9,  4,  1;
%e A055461    49, 36, 25, 16,  9,  4,  1;
%e A055461    64, 49, 36, 25, 16,  9,  4,  1;
%e A055461    81, 64, 49, 36, 25, 16,  9,  4,  1;
%e A055461   100, 81, 64, 49, 36, 25, 16,  9,  4,  1;
%e A055461   ...
%e A055461 For n = 7 the row sum is 49 + 36 + 25 + 16 + 9 + 4 + 1 = A000330(7) = 140.
%e A055461 The alternating row sum is 49 - 36 + 25 - 16 + 9 - 4 + 1 = A000217(7) = 28.
%e A055461 (End)
%p A055461 for n from 1 to 10 do
%p A055461   seq((n-k)^2, k=0..n-1)
%p A055461 od; # _Robert Israel_, Jan 18 2018
%t A055461 Table[Range[n,1,-1]^2,{n,20}]//Flatten (* _Harvey P. Dale_, Apr 17 2020 *)
%o A055461 (Magma) [(n-k)^2: k in [0..n-1], n in [1..15]]; // _G. C. Greubel_, Jan 31 2024
%o A055461 (SageMath) flatten([[(n-k)^2 for k in range(n)] for n in range(1,16)]) # _G. C. Greubel_, Jan 31 2024
%Y A055461 Cf. A000290, A000292, A004736, A129371, A194274.
%Y A055461 Cf. A000217 (alternating row sums), A000330 (row sums).
%K A055461 easy,nonn,tabl
%O A055461 1,2
%A A055461 _Henry Bottomley_, Jun 26 2000