A207616 Triangle of coefficients of polynomials u(n,x) jointly generated with A207617; see the Formula section.

1, 2, 4, 1, 7, 4, 11, 11, 1, 16, 25, 6, 22, 50, 22, 1, 29, 91, 63, 8, 37, 154, 154, 37, 1, 46, 246, 336, 129, 10, 56, 375, 672, 375, 56, 1, 67, 550, 1254, 957, 231, 12, 79, 781, 2211, 2211, 781, 79, 1, 92, 1079, 3718, 4719, 2288, 377, 14, 106, 1456, 6006

Offset: 1

Author

Clark Kimberling, Feb 20 2012

Keywords

• nonn
• tabf
• easy

Comments

With offset 0, equals the stretched Riordan array ((1 - z + z^2)/(1 - z)^3, z^2/(1 - z)^2) in the notation of Corsani et al., Section 2. - Peter Bala, Dec 31 2015

Examples

			First five rows:
   1
   2
   4  1
   7  4
  11 11  1
		

Links

• C. Corsani, D. Merlini, and R. Sprugnoli, Left-inversion of combinatorial sums, Discrete Mathematics, 180 (1998) 107-122.

Crossrefs

Cf. A207617, A208510, A000124 (column 1).

Programs

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_] := x*u[n - 1, 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[%]    (* A207616 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207617 *)

Formula

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