A208330 Triangle of coefficients of polynomials u(n,x) jointly generated with A208331; see the Formula section.

1, 1, 1, 1, 2, 3, 1, 3, 9, 5, 1, 4, 18, 20, 11, 1, 5, 30, 50, 55, 21, 1, 6, 45, 100, 165, 126, 43, 1, 7, 63, 175, 385, 441, 301, 85, 1, 8, 84, 280, 770, 1176, 1204, 680, 171, 1, 9, 108, 420, 1386, 2646, 3612, 3060, 1539, 341, 1, 10, 135, 600, 2310, 5292, 9030

Offset: 1

Author

Clark Kimberling, Feb 26 2012

Keywords

• nonn
• tabl

Comments

Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012

Examples

			First five rows:
1
1...1
1...2...3
1...3...9....5
1...4...18...20...11
First five polynomials u(n,x):
1, 1 + x, 1 + 2x + 3x^2, 1 + 3x + 9x^2 + 5x^3, 1 + 4x + 18x^2 + 20x^3 + 11x^4.
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins :
1
1, 0
1, 1, 0
1, 2, 3, 0
1, 3, 9, 5, 0
1, 4, 18, 20, 11, 0
1, 5, 30, 50, 55, 21, 0. - _Philippe Deléham_, Mar 18 2012
		

Crossrefs

Cf. A208331, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
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[%]  (* A208330 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208331 *)

Formula

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

v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),

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

T(n,k) = A001045(k+1)*binomial(n-1,k). - Philippe Deléham, Mar 18 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) - T(n-2,k-1) + 2*T(n-2,k-2), T(1,0) = T(2,0) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012