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

1,5

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

Links

Table of n, a(n) for n=1..62.

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

Example

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

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 *)

Crossrefs

Cf. A208331, A208510.

Sequence in context: A244490 A133935 A139633 * A152440 A134319 A135091

Adjacent sequences: A208327 A208328 A208329 * A208331 A208332 A208333

Keyword

nonn,tabl

Author

Clark Kimberling, Feb 26 2012

Status

approved