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

1,2

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

Links

Table of n, a(n) for n=1..59.

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

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.

Example

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

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 *)

Crossrefs

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

Sequence in context: A374116 A256107 A207610 * A105552 A112852 A121531

Adjacent sequences: A207613 A207614 A207615 * A207617 A207618 A207619

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Feb 20 2012

Status

approved