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A209567
Triangle of coefficients of polynomials u(n,x) jointly generated with A209568; see the Formula section.
3
1, 1, 1, 2, 3, 1, 3, 7, 6, 1, 4, 13, 18, 10, 1, 5, 21, 41, 39, 15, 1, 6, 31, 79, 108, 75, 21, 1, 7, 43, 136, 245, 250, 132, 28, 1, 8, 57, 216, 486, 661, 524, 217, 36, 1, 9, 73, 323, 875, 1497, 1601, 1015, 338, 45, 1, 10, 91, 461, 1464, 3031, 4109, 3556, 1844
OFFSET
1,4
COMMENTS
For n>1, row n begins with n and ends with 1.
For n>1, penultimate number in row n is (n-1)st triangular number.
Alternating row sums: 1,0,0,1,0,0,1,0,0,1,0,0,...
For a discussion and guide to related arrays, see A208510.
FORMULA
u(n,x)=x*u(n-1,x)+v(n-1,x),
v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
EXAMPLE
First five rows:
1
1...1
2...3....1
3...7....6....1
4...13...18...10...1
First three polynomials v(n,x): 1, 1 + x, 2 + 3x + x^2.
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%] (* A209567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A209568 *)
CROSSREFS
Sequence in context: A236376 A063967 A059397 * A208338 A236918 A152821
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Mar 10 2012
STATUS
approved