A102662 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).

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, 1, 15, 83, 211, 255, 143, 35, 3, 1, 17, 111, 353, 579, 489, 207, 41, 3, 1, 19, 143, 547, 1143, 1323, 839, 283, 47, 3, 1, 21, 179, 801, 2043, 3045, 2651, 1329, 371, 53, 3, 1, 23, 219

Offset: 1

Author

Lambert Klasen (lambert.klasen(AT)gmx.net) and Gary W. Adamson, Feb 03 2005

Keywords

• nonn
• tabl
• changed

Comments

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.

Examples

			Triangle begins:
  1
  1 3
  1 5 3
  1 7 11 3
  1 9 23 17 3
		

References

• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Links

• Reinhard Zumkeller, Rows n=0..149 of triangle, flattened
• Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 37.

Crossrefs

Cf. A038763, A048654, A008288.

Programs

Haskell

a102662 n k = a102662_tabl !! n !! k
a102662_row n = a102662_tabl !! n
a102662_tabl = [1] : [1,3] : f [1] [1,3] where
   f xs ys = zs : f ys zs where
     zs = zipWith (+) ([0] ++ xs ++ [0]) $
                      zipWith (+) ([0] ++ ys) (ys ++ [0])
-- Reinhard Zumkeller, Feb 23 2012

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + v[n - 1, x]
v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x] + 1
Table[Factor[u[n, x]], {n, 1, z}]
Table[Factor[v[n, x]], {n, 1, z}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A207624 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A102662 *)
(* Clark Kimberling, Feb 20 2012 *)

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,","))

Formula

From Clark Kimberling, Feb 20 2012: (Start)

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

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

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