A055461 - OEIS

%I #59 Feb 01 2024 04:19:28

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

%H Robert Israel, <a href="/A055461/b055461.txt">Rows n = 1..141, flattened</a>

%F a(n) = A004736(n)^2.

%F Sum_{k=0..n-1} T(n, k) = A000330(n) (row sums). - _Michel Marcus_, Dec 31 2012

%F G.f. as triangle: x*(1+x)/((1-x*y)*(1-x)^3). - _Robert Israel_, Jan 18 2018

%F Sum_{k=0..n-1} (-1)^k*T(n, k) = A000217(n) (alternating row sums). - _Omar E. Pol_, Jan 24 2014

%F From _G. C. Greubel_, Jan 31 2024: (Start)

%F T(2*n-1, n-1) = A000290(n).

%F Sum_{k=0..floor(n/2)} T(n-k, k) = A000292(n).

%F Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A194274(n).

%F Sum_{k=0..floor(n/2)} T(n, k) = A129371(n). (End)

%e From _Omar E. Pol_, Jan 26 2014: (Start)

%e Triangle begins:

%e     1;

%e     4,  1;

%e     9,  4,  1;

%e    16,  9,  4,  1;

%e    25, 16,  9,  4,  1;

%e    36, 25, 16,  9,  4,  1;

%e    49, 36, 25, 16,  9,  4,  1;

%e    64, 49, 36, 25, 16,  9,  4,  1;

%e    81, 64, 49, 36, 25, 16,  9,  4,  1;

%e   100, 81, 64, 49, 36, 25, 16,  9,  4,  1;

%e   ...

%e For n = 7 the row sum is 49 + 36 + 25 + 16 + 9 + 4 + 1 = A000330(7) = 140.

%e The alternating row sum is 49 - 36 + 25 - 16 + 9 - 4 + 1 = A000217(7) = 28.

%e (End)

%p for n from 1 to 10 do

%p   seq((n-k)^2, k=0..n-1)

%p od; # _Robert Israel_, Jan 18 2018

%t Table[Range[n,1,-1]^2,{n,20}]//Flatten (* _Harvey P. Dale_, Apr 17 2020 *)

%o (Magma) [(n-k)^2: k in [0..n-1], n in [1..15]]; // _G. C. Greubel_, Jan 31 2024

%o (SageMath) flatten([[(n-k)^2 for k in range(n)] for n in range(1,16)]) # _G. C. Greubel_, Jan 31 2024

%Y Cf. A000290, A000292, A004736, A129371, A194274.

%Y Cf. A000217 (alternating row sums), A000330 (row sums).

%K easy,nonn,tabl

%O 1,2

%A _Henry Bottomley_, Jun 26 2000