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

1,2

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

Links

Table of n, a(n) for n=1..53.

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

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

Example

First five rows:
   1
   2
   5  1
  11  4
  23 13  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 + 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 *)

Crossrefs

Cf. A207630, A208510, A083329 (column 1).

Sequence in context: A120235 A323411 A089618 * A207614 A156067 A352577

Adjacent sequences: A207626 A207627 A207628 * A207630 A207631 A207632

Keyword

nonn,tabf,easy

Author

Clark Kimberling, Feb 23 2012

Status

approved