A143596 - OEIS

%I #18 Sep 07 2024 16:04:07

%S 1,2,2,2,3,6,10,6,3,4,12,28,32,28,12,4,5,20,60,100,126,100,60,20,5,6,

%T 30,110,240,396,452,396,240,110,30,6,7,42,182,490,1001,1484,1716,1484,

%U 1001,490,182,42,7,8,56,280,896,2184,3976,5720,6400,5720,3976,2184,896,280

%N Triangle read by rows: T(n,k) is the coefficient of x^k in p(x,n) = ((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x), 0 <= k <= 2*n-1.

%C Row sums are:{1, 6, 28, 120, 496, 2016, 8128, 32640, 130816, 523776}.

%H Andrew Howroyd, <a href="/A143596/b143596.txt">Table of n, a(n) for n = 1..2500</a> (first 50 rows)

%F T(n,k) = [x^k] ((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x).

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

%F G.f.: x/(1 - 2*(1 + y + y^2)*x + (1 + y)^2*(1 + y^2)*x^2). - _Andrew Howroyd_, Sep 06 2024

%e {1},

%e {2, 2, 2},

%e {3, 6, 10, 6, 3},

%e {4, 12, 28, 32, 28, 12, 4},

%e {5, 20, 60, 100, 126, 100, 60, 20, 5},

%e {6, 30, 110, 240, 396, 452, 396, 240, 110, 30, 6},

%e {7, 42, 182, 490, 1001, 1484, 1716, 1484, 1001, 490, 182, 42, 7},

%e {8, 56, 280, 896, 2184, 3976, 5720, 6400, 5720, 3976, 2184, 896, 280, 56, 8},

%t Clear[p, x, n, m]; p[x_, n_] = ((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x); Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}]; Flatten[%]

%o (PARI) row(n)=Vecrev(((x + 1)^(2*n) - (x^2 + 1)^n)/(2*x)) \\ _Andrew Howroyd_, Sep 06 2024

%Y Cf. A006516 (row sums).

%K nonn,tabf

%O 1,2

%A _Roger L. Bagula_ and _Gary W. Adamson_, Oct 25 2008