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A208336
Triangle of coefficients of polynomials u(n,x) jointly generated with A208337; see the Formula section.
5
1, 1, 1, 1, 2, 2, 1, 3, 5, 3, 1, 4, 9, 10, 5, 1, 5, 14, 22, 20, 8, 1, 6, 20, 40, 51, 38, 13, 1, 7, 27, 65, 105, 111, 71, 21, 1, 8, 35, 98, 190, 256, 233, 130, 34, 1, 9, 44, 140, 315, 511, 594, 474, 235, 55, 1, 10, 54, 192, 490, 924, 1295, 1324, 942, 420, 89, 1, 11
OFFSET
1,5
COMMENTS
coef. of x^(n-1) in u(n,x): A000045(n), Fibonacci numbers
coef. of x^(n-1) in v(n,x): A000045(n+1)
row sums, u(n,1): A000129
row sums, v(n,1): A001333
alternating row sums, u(n,-1): 1,0,1,0,1,0,1,0,1,0,...
alternating row sums, v(n,-1): 1,-1,1,-1,1,-1,1,-1,...
FORMULA
u(n,x)=u(n-1,x)+x*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
T(n,k) = A038137(n-1,k). - Philippe Deléham, Apr 05 2012
EXAMPLE
First five rows:
1
1...1
1...2...2
1...3...5...3
1...4...9...10...5
First five polynomials u(n,x):
1
1 + x
1 + 2x + 2x^2
1 + 3x + 5x^2 + 3x^3
1 + 4x + 9x^2 + 10x^3 + 5x^4
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_] := (x + 1)*u[n - 1, x] + x*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[%] (* A208336 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A208337 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* u row sums *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* v row sums *)
Table[u[n, x] /. x -> -1, {n, 1, z}](* u alt. row sums *)
Table[v[n, x] /. x -> -1, {n, 1, z}](* v alt. row sums *)
CROSSREFS
Apart from offsets the same as A038137.
Cf. A208337.
Sequence in context: A239830 A140767 A060850 * A038137 A073133 A106179
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Feb 26 2012
STATUS
approved