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A208331
Triangle of coefficients of polynomials v(n,x) jointly generated with A208330; see the Formula section.
3
1, 1, 3, 1, 6, 5, 1, 9, 15, 11, 1, 12, 30, 44, 21, 1, 15, 50, 110, 105, 43, 1, 18, 75, 220, 315, 258, 85, 1, 21, 105, 385, 735, 903, 595, 171, 1, 24, 140, 616, 1470, 2408, 2380, 1368, 341, 1, 27, 180, 924, 2646, 5418, 7140, 6156, 3069, 683, 1, 30, 225
OFFSET
1,3
COMMENTS
Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012
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(n+2)*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) = 1, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - Philippe Deléham, Mar 18 2012
EXAMPLE
First five rows:
1
1...3
1...6...5
1...9...15...11
1...12...30...44...21
First five polynomials u(n,x):
1, 1 + 3x, 1 + 6x + 5x^2, 1 + 9x + 15x^2 + 11x^3, 1+12x + 30x^2 + 44x^3 + 21x^4.
(1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, ...) begins :
1
1, 0
1, 3, 0
1, 6, 5, 0
1, 9, 15, 11, 0
1, 12, 30, 44, 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. A208330.
Sequence in context: A308948 A225246 A116666 * A061702 A112351 A143858
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved