A395065 - OEIS

%I #28 May 12 2026 16:21:09

%S 1,1,1,1,4,3,1,9,24,20,1,16,90,216,189,1,25,240,1120,2560,2304,1,36,

%T 525,4000,16875,37500,34375,1,49,1008,11340,75600,299376,653184,

%U 606528,1,64,1764,27440,264110,1613472,6117748,13176688,12353145

%N Triangle read by rows: coefficients of (1+(2*n-1)*x)*(1+(n-1)*x)^(n-1).

%e The first few rows are:

%e   1;

%e   1,  1;

%e   1,  4,  3;

%e   1,  9, 24,  20;

%e   1, 16, 90, 216, 189;

%e   ...

%t T[n_]:=CoefficientList[ (1 + (2 n - 1) x) * (1 + (n - 1) x)^(n - 1),x];Array[T,9,0]//Flatten (* _James C. McMahon_, May 12 2026 *)

%o (SageMath)

%o x = polygen(ZZ, 'x')

%o def row(n):

%o     if n == 0: return [1]

%o     return list((1+(2*n-1)*x)*(1+(n-1)*x)**(n-1))

%o (PARI) row(n) = Vec((1+(2*n-1)*x)*(1+(n-1)*x)^(n-1)); \\ _Michel Marcus_, May 09 2026

%Y Row sums are A013499.

%Y Main diagonal gives A176043.

%Y Columns k=0-1 give: A000012, A000290.

%Y Similar to A395611, A071207, A243594, A395505.

%K nonn,tabl,easy

%O 0,5

%A _F. Chapoton_, May 08 2026