A207629 Triangle of coefficients of polynomials u(n,x) jointly generated with A207630; see the Formula section.

1, 2, 5, 1, 11, 4, 23, 13, 1, 47, 37, 6, 95, 97, 25, 1, 191, 241, 87, 8, 383, 577, 271, 41, 1, 767, 1345, 783, 169, 10, 1535, 3073, 2143, 609, 61, 1, 3071, 6913, 5631, 2001, 291, 12, 6143, 15361, 14335, 6145, 1191, 85, 1, 12287, 33793, 35583, 17921

Offset: 1

Author

Clark Kimberling, Feb 23 2012

Keywords

• nonn
• tabf
• easy

Comments

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

Examples

			First five rows:
   1
   2
   5  1
  11  4
  23 13  1
		

Links

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

Crossrefs

Cf. A207630, A208510, A083329 (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 + 1)*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[%]    (* A207629 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A207630 *)

Formula

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

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

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