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Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
3

%I #13 Aug 03 2014 14:27:17

%S 1,1,3,1,5,3,1,7,11,3,1,9,23,17,3,1,11,39,51,23,3,1,13,59,113,91,29,3,

%T 1,15,83,211,255,143,35,3,1,17,111,353,579,489,207,41,3,1,19,143,547,

%U 1143,1323,839,283,47,3,1,21,179,801,2043,3045,2651,1329,371,53,3,1,23,219

%N Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).

%C Generalization of A008288 (use initial terms 1,1,3). Triangle seen as lower triangular matrix: The absolute values of the coefficients of the characteristic polynomials of the n X n matrix are the (n+1)th row of A038763. Row sums give A048654.

%D Boris A. Bondarenko, "Generalized Pascal Triangles and Pyramids" Fibonacci Association, 1993, p. 37

%H Reinhard Zumkeller, <a href="/A102662/b102662.txt">Rows n=0..149 of triangle, flattened</a>

%F A102662=v and A207624=u, defined together as follows:

%F u(n,x)=u(n-1,x)+v(n-1,x), v(n,x)=2x*u(n-1,x)+x*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1; see the Mathematica section.

%F [From Clark Kimberling, Feb 20 2012]

%e Triangle begins:

%e 1

%e 1 3

%e 1 5 3

%e 1 7 11 3

%e 1 9 23 17 3

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

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

%t v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1

%t Table[Factor[u[n, x]], {n, 1, z}]

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

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

%t TableForm[cu]

%t Flatten[%] (* A207624 *)

%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[%] (* A102662 *)

%t (* _Clark Kimberling_, Feb 20 2012 *)

%o (PARI) T(k,r)=if(r>k,0,if(k==1,1,if(k==2,if(r==1,1,3),if(r==1,1,if(r==k,3,T(k-1,r-1)+T(k-1,r)+T(k-2,r-1)))))) BM(n) = M=matrix(n,n);for(i=1,n, for(j=1,n,M[i,j]=T(i,j)));M M=BM(10) for(i=1,10,s=0;for(j=1,i,s+=M[i,j]);print1(s,","))

%o (Haskell)

%o a102662 n k = a102662_tabl !! n !! k

%o a102662_row n = a102662_tabl !! n

%o a102662_tabl = [1] : [1,3] : f [1] [1,3] where

%o f xs ys = zs : f ys zs where

%o zs = zipWith (+) ([0] ++ xs ++ [0]) $

%o zipWith (+) ([0] ++ ys) (ys ++ [0])

%o -- _Reinhard Zumkeller_, Feb 23 2012

%Y Cf. A038763, A048654, A008288.

%K nonn,tabl

%O 1,3

%A Lambert Klasen (lambert.klasen(AT)gmx.net) and _Gary W. Adamson_, Feb 03 2005