A202390 Triangle T(n,k), read by rows, given by (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 1, 1, 1, 3, 2, 1, 6, 8, 3, 1, 10, 21, 17, 5, 1, 15, 45, 58, 35, 8, 1, 21, 85, 154, 144, 68, 13, 1, 28, 147, 350, 452, 330, 129, 21, 1, 36, 238, 714, 1195, 1198, 719, 239, 34, 1, 45, 366, 1344, 2799, 3611, 2959, 1506, 436, 55

Offset: 0

Author

Philippe Deléham, Dec 18 2011

Keywords

• nonn
• tabl
• changed

Comments

T(n,n) = Fibonacci(n+1) = A000045(n+1).

A202390 is jointly generated with A208340 as an array of coefficients of polynomials v(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+x*v(n-1)x and v(n,x)=(x+1)*u(n-1,x)+(x+1)v(n-1,x). The alternating row sums of A202390, and also A208340, are 0 except for the first one. See the Mathematica section. - Clark Kimberling, Feb 27 2012

Examples

			Triangle begins:
  1
  1, 1
  1, 3, 2
  1, 6, 8, 3
  1, 10, 21, 17, 5
  1, 15, 45, 58, 35, 8
  1, 21, 85, 154, 144, 68, 13
  1, 28, 147, 350, 452, 330, 129, 21
		

Crossrefs

Cf. A000012, A000217, A051744, A000045, A123585, A208340.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A202390 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208340 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (*row sums*)
Table[u[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)

Formula

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k) with T(0,0) = T(1,0) = T(1,1) = 1 and T(n,k) = 0 if k<0 or if n

G.f.: (1-x)/(1-(2+y)*x+(1-y^2)*x^2).

Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A108411(n), A000007(n), A000012(n), A025192(n), A122558(n) for x = -2, -1, 0, 1, 2 respectively.

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