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A103218 Triangle read by rows: T(n, k) = (2*k+1)*(n+1-k)^2. 1

%I

%S 1,4,3,9,12,5,16,27,20,7,25,48,45,28,9,36,75,80,63,36,11,49,108,125,

%T 112,81,44,13,64,147,180,175,144,99,52,15,81,192,245,252,225,176,117,

%U 60,17,100,243,320,343,324,275,208,135,68,19,121,300,405,448,441,396,325,240

%N Triangle read by rows: T(n, k) = (2*k+1)*(n+1-k)^2.

%C The triangle is generated from the product A * B of the infinite lower triangular matrix A =

%C 1 0 0 0...

%C 3 1 0 0...

%C 5 3 1 0...

%C 7 5 3 1...

%C ... and B =

%C 1 0 0 0...

%C 1 3 0 0...

%C 1 3 5 0...

%C 1 3 5 7...

%C ...

%e Triangle begins:

%e 1,

%e 4,3,

%e 9,12,5,

%e 16,27,20,7,

%e 25,48,45,28,9,

%t T[n_, k_] := (2*k + 1)*(n + 1 - k)^2; Flatten[ Table[ T[n, k], {n, 0, 10}, {k, 0, n}]] (* _Robert G. Wilson v_, Feb 10 2005 *)

%o (PARI) T(n, k) = (2*k+1)*(n+1-k)^2; for(i=0,10, for(j=0,i,print1(T(i,j),","));print())

%Y Row sums give A002412 (hexagonal pyramidal numbers).

%Y T(n, 0)=A000290(n+1) (the squares);

%Y T(n, 1)=3*n^2=A033428(n);

%Y T(n, 2)=5*n^2=A033429(n+1);

%Y T(n, 3)=7*n^2=A033582(n+2);

%Y Cf. A103219 (product B*A), A002412, A000290.

%K nonn,tabl

%O 0,2

%A Lambert Klasen (lambert.klasen(AT)gmx.de) and _Gary W. Adamson_, Jan 25 2005

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Last modified February 29 02:35 EST 2020. Contains 332353 sequences. (Running on oeis4.)