A209577 Triangle of coefficients of polynomials u(n,x) jointly generated with A209578; see the Formula section.

1, 1, 1, 3, 2, 1, 5, 6, 3, 1, 9, 13, 10, 4, 1, 15, 28, 26, 15, 5, 1, 25, 56, 64, 45, 21, 6, 1, 41, 109, 146, 124, 71, 28, 7, 1, 67, 206, 319, 315, 216, 105, 36, 8, 1, 109, 382, 671, 758, 602, 349, 148, 45, 9, 1, 177, 697, 1372, 1744, 1576, 1056, 533, 201, 55, 10

Offset

1,4

Comments

For a discussion and guide to related arrays, see A208510.

Links

Table of n, a(n) for n=1..65.

Formula

u(n,x) = 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.

The coefficients in the triangle seem to be T(n,m) = sum(k=0,n-m,2 * binomial(m+k, m)*binomial(k, n-k-m) - sum(i=0, n-m-k, binomial(m+k-1,k)*binomial(k,n-m-i-k))) by using the PARI syntax. - Thomas Baruchel, Jun 03 2018

Example

First five rows:
  1;
  1,  1;
  3,  2,  1;
  5,  6,  3,  1;
  9, 13, 10,  4,  1;
First three polynomials v(n,x): 1, 1 + x, 3 + 2x + x^2.

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x] + 1;
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[%]   (* A209577 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A209578 *)

Crossrefs

Cf. A209578, A208510.

Sequence in context: A132970 A192022 A208608 * A139377 A368607 A138483

Adjacent sequences: A209574 A209575 A209576 * A209578 A209579 A209580

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 11 2012

Status

approved