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A125092 Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).

%I A125092 #15 Feb 20 2023 10:38:10
%S A125092 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 A125092 375,216,49,1,28,189,560,875,756,343,64,1,32,252,896,1750,2016,1372,
%U A125092 512,81,1,36,324,1344,3150,4536,4116,2304,729,100,1,40,405,1920,5250,9072
%N A125092 Triangle read by rows: T(n,k) = (k+1)^2*binomial(n,k) (0 <= k <= n).
%C A125092 Binomial transform of the infinite diagonal matrix (1,4,9,16,...).
%C A125092 Sum of entries in row n = (n+1)*(n+4)*2^(n-2) = A001793(n+1).
%H A125092 Harvey P. Dale, <a href="/A125092/b125092.txt">Table of n, a(n) for n = 0..1000</a>
%e A125092 First few rows of the triangle:
%e A125092   1;
%e A125092   1,   4;
%e A125092   1,   8,   9;
%e A125092   1,  12,  27,  16;
%e A125092   1,  16,  54,  64,  25;
%e A125092   1,  20,  90, 160, 125,  36;
%e A125092   ...
%p A125092 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 A125092 Table[(k+1)^2 Binomial[n,k],{n,0,10},{k,0,n}]//Flatten (* _Harvey P. Dale_, Feb 20 2023 *)
%Y A125092 Cf. A001793.
%K A125092 nonn,tabl
%O A125092 0,3
%A A125092 _Gary W. Adamson_, Nov 19 2006
%E A125092 Edited by _N. J. A. Sloane_, Nov 29 2006