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Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
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%I #15 Feb 20 2023 10:38:10

%S 1,1,4,1,8,9,1,12,27,16,1,16,54,64,25,1,20,90,160,125,36,1,24,135,320,

%T 375,216,49,1,28,189,560,875,756,343,64,1,32,252,896,1750,2016,1372,

%U 512,81,1,36,324,1344,3150,4536,4116,2304,729,100,1,40,405,1920,5250,9072

%N Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).

%C Binomial transform of the infinite diagonal matrix (1,4,9,16,...).

%C Sum of entries in row n = (n+1)*(n+4)*2^(n-2) = A001793(n+1).

%H Harvey P. Dale, <a href="/A125092/b125092.txt">Table of n, a(n) for n = 0..1000</a>

%e First few rows of the triangle:

%e 1;

%e 1, 4;

%e 1, 8, 9;

%e 1, 12, 27, 16;

%e 1, 16, 54, 64, 25;

%e 1, 20, 90, 160, 125, 36;

%e ...

%p T:=(n,k)->(k+1)^2*binomial(n,k): for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%t Table[(k+1)^2 Binomial[n,k],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Feb 20 2023 *)

%Y Cf. A001793.

%K nonn,tabl

%O 0,3

%A _Gary W. Adamson_, Nov 19 2006

%E Edited by _N. J. A. Sloane_, Nov 29 2006