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A207636
Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section.
3
1, 3, 2, 6, 7, 2, 12, 20, 11, 2, 24, 52, 42, 15, 2, 48, 128, 136, 72, 19, 2, 96, 304, 400, 280, 110, 23, 2, 192, 704, 1104, 960, 500, 156, 27, 2, 384, 1600, 2912, 3024, 1960, 812, 210, 31, 2, 768, 3584, 7424, 8960, 6944, 3584, 1232, 272, 35, 2, 1536, 7936
OFFSET
1,2
COMMENTS
As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012
FORMULA
u(n,x) = u(n-1,x) + v(n-1,x),
v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x) + 1,
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k), 0 <= k <= n:
T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 3, T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
G.f.: (1+x+y*x)/(1-2*x-y*x).
Sum_{k=0..n} T(n,k)*x^k = A003945(n), |A084244(n)|, A189274(n) for x = 0, 1, 3 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A040000(n), |A084244(n)|, A128625(n) for x = 0, 1, 2 respectively. (End)
EXAMPLE
First five rows:
1;
3, 2;
6, 7, 2;
12, 20, 11, 2;
24, 52, 42, 15, 2;
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] + (x + 1)*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[%] (* A207635 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A207636 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 24 2012
STATUS
approved