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T(n, k) = -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2). Triangle read by rows, T(n, k) for 1 <= k <= n.
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%I #21 Dec 21 2023 10:22:26

%S 1,2,2,3,6,3,4,12,13,4,5,20,34,24,5,6,30,70,80,40,6,7,42,125,200,166,

%T 62,7,8,56,203,420,496,314,91,8,9,72,308,784,1211,1106,553,128,9,10,

%U 90,444,1344,2576,3108,2269,920,174,10,11,110,615,2160,4956,7476,7274,4352,1461,230,11

%N T(n, k) = -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2). Triangle read by rows, T(n, k) for 1 <= k <= n.

%C Previous name: Triangle of coefficients of polynomials v(n,x) jointly generated with A210217.

%C For a discussion and guide to related arrays, see A208510.

%F First and last term in row n: n.

%F Column 2: n*(n-1).

%F Column 3: A016061.

%F Column 4: A112742.

%F Row sums: -1+(even-indexed Fibonacci numbers).

%F Periodic alternating row sums: 1,0,0,1,0,0,1,0,0,...

%F u(n,x)=x*u(n-1,x)+v(n-1,x)+1,

%F v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%F T(n,k) = Sum_{j=1..n-k+1} binomial(2*j+k-2,k-1). - _Detlef Meya_, Dec 05 2023

%e First five rows:

%e 1

%e 2...2

%e 3...6....3

%e 4...12...13...4

%e 5...20...34...24...5

%e First three polynomials v(n,x): 1, 2 + 2x , 3 + 6x + 3x^2.

%p T := (n,k) -> -binomial(2*n-k+2, k+1)*hypergeom([2*n-k+3, 1], [k+2], 2):

%p seq(seq(simplify(T(n,k)), k=1..n), n=1..10); # _Peter Luschny_, Oct 31 2019

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;

%t v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A210219 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A210220 *)

%t (* alternate program *)

%t T[n_,k_]:=Sum[Binomial[2*j+k-2,k-1],{j,1,n-k+1}];Flatten[Table[T[n,k],{n,1,11},{k,1,n}]] (* _Detlef Meya_, Dec 05 2023 *)

%Y Cf. A210217, A210219, A208510, A016061, A112742.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 19 2012

%E New name from _Peter Luschny_, Oct 31 2019